Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains
|
|
- Eustace Kerry Horton
- 5 years ago
- Views:
Transcription
1 Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains Matteo Bonforte Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco Madrid, Spain BCAM Scientific Seminar BCAM Basque Center for Applied Mathematics Bilbao, Spain, May 30, 207
2 References References: [BV] M. B., J. L. VÁZQUEZ, A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains. Arch. Rat. Mech. Anal. (205). [BV2] M. B., J. L. VÁZQUEZ, Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains Part I. Existence, Uniqueness and Upper Bounds Nonlin. Anal. TMA (206). [BSV] M. B., Y. SIRE, J. L. VÁZQUEZ, Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains. Discr. Cont. Dyn. Sys. (205). [BFR] M. B., A. FIGALLI, X. ROS-OTON, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. To appear in Comm. Pure Appl. Math (207). [BFV] M. B., A. FIGALLI, J. L. VÁZQUEZ, Sharp boundary estimates and higher regularity for nonlocal porous medium-type equations in bounded domains. Preprint (206). A talk more focussed on the first part is available online:
3 Summary Outline of the talk Introduction The abstract setup of the problem Some important examples About Spectral Kernels Basic Theory The Dual problem Existence and uniqueness First set of estimates Sharp Boundary Behaviour Upper Boundary Estimates Infinite Speed of Propagation Lower Boundary Estimates Harnack Inequalities Numerics Regularity Estimates
4 Homogeneous Dirichlet Problem for Fractional Nonlinear Degenerate Diffusion Equations u t + L F(u) = 0, in (0, + ) Ω (HDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. where: Ω R N is a bounded domain with smooth boundary and N. The linear operator L will be: sub-markovian operator densely defined in L (Ω). A wide class of linear operators fall in this class: all fractional Laplacians on domains. The most studied nonlinearity is F(u) = u m u, with m >. We deal with Degenerate diffusion of Porous Medium type. More general classes of degenerate nonlinearities F are allowed. The homogeneous boundary condition is posed on the lateral boundary, which may take different forms, depending on the particular choice of the operator L.
5 Homogeneous Dirichlet Problem for Fractional Nonlinear Degenerate Diffusion Equations u t + L F(u) = 0, in (0, + ) Ω (HDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. where: Ω R N is a bounded domain with smooth boundary and N. The linear operator L will be: sub-markovian operator densely defined in L (Ω). A wide class of linear operators fall in this class: all fractional Laplacians on domains. The most studied nonlinearity is F(u) = u m u, with m >. We deal with Degenerate diffusion of Porous Medium type. More general classes of degenerate nonlinearities F are allowed. The homogeneous boundary condition is posed on the lateral boundary, which may take different forms, depending on the particular choice of the operator L.
6 Homogeneous Dirichlet Problem for Fractional Nonlinear Degenerate Diffusion Equations u t + L F(u) = 0, in (0, + ) Ω (HDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. where: Ω R N is a bounded domain with smooth boundary and N. The linear operator L will be: sub-markovian operator densely defined in L (Ω). A wide class of linear operators fall in this class: all fractional Laplacians on domains. The most studied nonlinearity is F(u) = u m u, with m >. We deal with Degenerate diffusion of Porous Medium type. More general classes of degenerate nonlinearities F are allowed. The homogeneous boundary condition is posed on the lateral boundary, which may take different forms, depending on the particular choice of the operator L.
7 Homogeneous Dirichlet Problem for Fractional Nonlinear Degenerate Diffusion Equations u t + L F(u) = 0, in (0, + ) Ω (HDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. where: Ω R N is a bounded domain with smooth boundary and N. The linear operator L will be: sub-markovian operator densely defined in L (Ω). A wide class of linear operators fall in this class: all fractional Laplacians on domains. The most studied nonlinearity is F(u) = u m u, with m >. We deal with Degenerate diffusion of Porous Medium type. More general classes of degenerate nonlinearities F are allowed. The homogeneous boundary condition is posed on the lateral boundary, which may take different forms, depending on the particular choice of the operator L.
8 About the operator L The linear operator L : dom(a) L (Ω) L (Ω) is assumed to be densely defined and sub-markovian, more precisely satisfying (A) and (A2) below: (A) L is m-accretive on L (Ω), (A2) If 0 f then 0 e tl f, or equivalently, (A2 ) If β is a maximal monotone graph in R R with 0 β(0), u dom(l), Lu L p (Ω), p, v L p/(p ) (Ω), v(x) β(u(x)) a.e, then v(x)lu(x) dx 0 Ω Remark. These assumptions are needed for existence (and uniqueness) of semigroup (mild) solutions for the nonlinear equation u t = LF(u), through a remarkable variant of the celebrated Crandall-Liggett theorem, as done by Benilan, Crandall and Pierre: M. G. Crandall, T.M. Liggett. Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (97) M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
9 About the operator L The linear operator L : dom(a) L (Ω) L (Ω) is assumed to be densely defined and sub-markovian, more precisely satisfying (A) and (A2) below: (A) L is m-accretive on L (Ω), (A2) If 0 f then 0 e tl f, or equivalently, (A2 ) If β is a maximal monotone graph in R R with 0 β(0), u dom(l), Lu L p (Ω), p, v L p/(p ) (Ω), v(x) β(u(x)) a.e, then v(x)lu(x) dx 0 Ω Remark. These assumptions are needed for existence (and uniqueness) of semigroup (mild) solutions for the nonlinear equation u t = LF(u), through a remarkable variant of the celebrated Crandall-Liggett theorem, as done by Benilan, Crandall and Pierre: M. G. Crandall, T.M. Liggett. Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (97) M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
10 About the operator L The linear operator L : dom(a) L (Ω) L (Ω) is assumed to be densely defined and sub-markovian, more precisely satisfying (A) and (A2) below: (A) L is m-accretive on L (Ω), (A2) If 0 f then 0 e tl f, or equivalently, (A2 ) If β is a maximal monotone graph in R R with 0 β(0), u dom(l), Lu L p (Ω), p, v L p/(p ) (Ω), v(x) β(u(x)) a.e, then v(x)lu(x) dx 0 Ω Remark. These assumptions are needed for existence (and uniqueness) of semigroup (mild) solutions for the nonlinear equation u t = LF(u), through a remarkable variant of the celebrated Crandall-Liggett theorem, as done by Benilan, Crandall and Pierre: M. G. Crandall, T.M. Liggett. Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (97) M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
11 Assumption on the nonlinearity F Let F : R R be a continuous and non-decreasing function, with F(0) = 0. Moreover, it satisfies the condition: (N) F C (R \ {0}) and F/F Lip(R) and there exists µ 0, µ > 0 s.t. m = µ ( F F ) µ 0 = where F/F is understood to vanish if F(r) = F (r) = 0 or r = 0. The main example (treated in the rest of the talk) will be F(u) = u m u, with m >, µ 0 = µ = m m <. A simple variant is the combination of two powers: m behaviour when u and m 0 behaviour when u 0 Monotonicity estimates follow by (N): the following maps m 0 t t µ 0 F(u(t, x)) or t t m u(t, x) are nondecreasing in t > 0 for a.e. x Ω. P. Bénilan, M. G. Crandall. Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, suppl. to Amer. Jour. Math., (98). Pp M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
12 Assumption on the nonlinearity F Let F : R R be a continuous and non-decreasing function, with F(0) = 0. Moreover, it satisfies the condition: (N) F C (R \ {0}) and F/F Lip(R) and there exists µ 0, µ > 0 s.t. m = µ ( F F ) µ 0 = where F/F is understood to vanish if F(r) = F (r) = 0 or r = 0. The main example (treated in the rest of the talk) will be F(u) = u m u, with m >, µ 0 = µ = m m <. A simple variant is the combination of two powers: m behaviour when u and m 0 behaviour when u 0 Monotonicity estimates follow by (N): the following maps m 0 t t µ 0 F(u(t, x)) or t t m u(t, x) are nondecreasing in t > 0 for a.e. x Ω. P. Bénilan, M. G. Crandall. Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, suppl. to Amer. Jour. Math., (98). Pp M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
13 Assumption on the nonlinearity F Let F : R R be a continuous and non-decreasing function, with F(0) = 0. Moreover, it satisfies the condition: (N) F C (R \ {0}) and F/F Lip(R) and there exists µ 0, µ > 0 s.t. m = µ ( F F ) µ 0 = where F/F is understood to vanish if F(r) = F (r) = 0 or r = 0. The main example (treated in the rest of the talk) will be F(u) = u m u, with m >, µ 0 = µ = m m <. A simple variant is the combination of two powers: m behaviour when u and m 0 behaviour when u 0 Monotonicity estimates follow by (N): the following maps m 0 t t µ 0 F(u(t, x)) or t t m u(t, x) are nondecreasing in t > 0 for a.e. x Ω. P. Bénilan, M. G. Crandall. Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, suppl. to Amer. Jour. Math., (98). Pp M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
14 Assumption on the nonlinearity F Let F : R R be a continuous and non-decreasing function, with F(0) = 0. Moreover, it satisfies the condition: (N) F C (R \ {0}) and F/F Lip(R) and there exists µ 0, µ > 0 s.t. m = µ ( F F ) µ 0 = where F/F is understood to vanish if F(r) = F (r) = 0 or r = 0. The main example (treated in the rest of the talk) will be F(u) = u m u, with m >, µ 0 = µ = m m <. A simple variant is the combination of two powers: m behaviour when u and m 0 behaviour when u 0 Monotonicity estimates follow by (N): the following maps m 0 t t µ 0 F(u(t, x)) or t t m u(t, x) are nondecreasing in t > 0 for a.e. x Ω. P. Bénilan, M. G. Crandall. Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, suppl. to Amer. Jour. Math., (98). Pp M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
15 Assumption on the nonlinearity F Let F : R R be a continuous and non-decreasing function, with F(0) = 0. Moreover, it satisfies the condition: (N) F C (R \ {0}) and F/F Lip(R) and there exists µ 0, µ > 0 s.t. m = µ ( F F ) µ 0 = where F/F is understood to vanish if F(r) = F (r) = 0 or r = 0. The main example (treated in the rest of the talk) will be F(u) = u m u, with m >, µ 0 = µ = m m <. A simple variant is the combination of two powers: m behaviour when u and m 0 behaviour when u 0 Monotonicity estimates follow by (N): the following maps m 0 t t µ 0 F(u(t, x)) or t t m u(t, x) are nondecreasing in t > 0 for a.e. x Ω. P. Bénilan, M. G. Crandall. Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, suppl. to Amer. Jour. Math., (98). Pp M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22
16 Assumption on the inverse operator L Assumptions on the inverse of L We will assume that the operator L has an inverse L : L (Ω) L (Ω) with a kernel K such that L f (x) = K(x, y) f (y) dy, Ω and that satisfies (one of) the following estimates for some γ, s (0, ] and c i,ω > 0 (K) (K2) c 0,Ωδ γ (x) δ γ (y) K(x, y) 0 K(x, y) c,ω x y N 2s ( ) ( ) c,ω δ γ (x) δ γ x y N 2s x y (y) γ x y γ where δ γ (x) := dist(x, Ω) γ. When L has a first eigenfunction, (K) implies 0 Φ L (Ω). Moreover, (K2) implies that Φ dist(, Ω) γ = δ γ and we can rewrite (K2) as ( ) ( ) c,ω Φ(x) Φ(y) (K3) c 0,ΩΦ (x)φ (y) K(x, y) x x 0 N 2s x y γ x y γ
17 Assumption on the inverse operator L Assumptions on the inverse of L We will assume that the operator L has an inverse L : L (Ω) L (Ω) with a kernel K such that L f (x) = K(x, y) f (y) dy, Ω and that satisfies (one of) the following estimates for some γ, s (0, ] and c i,ω > 0 (K) (K2) c 0,Ωδ γ (x) δ γ (y) K(x, y) 0 K(x, y) c,ω x y N 2s ( ) ( ) c,ω δ γ (x) δ γ x y N 2s x y (y) γ x y γ where δ γ (x) := dist(x, Ω) γ. When L has a first eigenfunction, (K) implies 0 Φ L (Ω). Moreover, (K2) implies that Φ dist(, Ω) γ = δ γ and we can rewrite (K2) as ( ) ( ) c,ω Φ(x) Φ(y) (K3) c 0,ΩΦ (x)φ (y) K(x, y) x x 0 N 2s x y γ x y γ
18 Examples of operators L Reminder about the fractional Laplacian operator on R N We have several equivalent definitions for ( R N ) s : By means of Fourier Transform, (( R N ) s f ) (ξ) = ξ 2sˆf (ξ). This formula can be used for positive and negative values of s. 2 By means of an Hypersingular Kernel: if 0 < s <, we can use the representation ( R N ) s g(x) = c N,s P.V. where c N,s > 0 is a normalization constant. R N g(x) g(z) dz, x z N+2s 3 Spectral definition, in terms of the heat semigroup associated to the standard Laplacian operator: ( R N ) s g(x) = Γ( s) 0 ( e t R N g(x) g(x) ) dt t +s.
19 Examples of operators L Reminder about the fractional Laplacian operator on R N We have several equivalent definitions for ( R N ) s : By means of Fourier Transform, (( R N ) s f ) (ξ) = ξ 2sˆf (ξ). This formula can be used for positive and negative values of s. 2 By means of an Hypersingular Kernel: if 0 < s <, we can use the representation ( R N ) s g(x) = c N,s P.V. where c N,s > 0 is a normalization constant. R N g(x) g(z) dz, x z N+2s 3 Spectral definition, in terms of the heat semigroup associated to the standard Laplacian operator: ( R N ) s g(x) = Γ( s) 0 ( e t R N g(x) g(x) ) dt t +s.
20 Examples of operators L Reminder about the fractional Laplacian operator on R N We have several equivalent definitions for ( R N ) s : By means of Fourier Transform, (( R N ) s f ) (ξ) = ξ 2sˆf (ξ). This formula can be used for positive and negative values of s. 2 By means of an Hypersingular Kernel: if 0 < s <, we can use the representation ( R N ) s g(x) = c N,s P.V. where c N,s > 0 is a normalization constant. R N g(x) g(z) dz, x z N+2s 3 Spectral definition, in terms of the heat semigroup associated to the standard Laplacian operator: ( R N ) s g(x) = Γ( s) 0 ( e t R N g(x) g(x) ) dt t +s.
21 Examples of operators L The Spectral Fractional Laplacian operator (SFL) ( Ω) s g(x) = λ s j ĝ j φ j(x) = ( ) e t Ω dt g(x) g(x) Γ( s) t. +s j= 0 Ω is the classical Dirichlet Laplacian on the domain Ω EIGENVALUES: 0 < λ λ 2... λ j λ j+... and λ j j 2/N. EIGENFUNCTIONS: φ j are as smooth as the boundary of Ω allows, namely when Ω is C k, then φ j C (Ω) C k (Ω) for all k N. ĝ j = g(x)φ j(x) dx, with φ j L 2 (Ω) =. Lateral boundary conditions for the SFL Ω u(t, x) = 0, in (0, ) Ω. The Green function of SFL satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = γ
22 Examples of operators L The Spectral Fractional Laplacian operator (SFL) ( Ω) s g(x) = λ s j ĝ j φ j(x) = ( ) e t Ω dt g(x) g(x) Γ( s) t. +s j= 0 Ω is the classical Dirichlet Laplacian on the domain Ω EIGENVALUES: 0 < λ λ 2... λ j λ j+... and λ j j 2/N. EIGENFUNCTIONS: φ j are as smooth as the boundary of Ω allows, namely when Ω is C k, then φ j C (Ω) C k (Ω) for all k N. ĝ j = g(x)φ j(x) dx, with φ j L 2 (Ω) =. Lateral boundary conditions for the SFL Ω u(t, x) = 0, in (0, ) Ω. The Green function of SFL satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = γ
23 Examples of operators L Definition via the hypersingular kernel in R N, restricted to functions that are zero outside Ω. The (Restricted) Fractional Laplacian operator (RFL) ( Ω ) s g(x) g(z) g(x) = c N,s P.V. dz, with supp(g) Ω. R N x z N+2s where s (0, ) and c N,s > 0 is a normalization constant. ( Ω ) s is a self-adjoint operator on L 2 (Ω) with a discrete spectrum: EIGENVALUES: 0 < λ λ 2... λ j λ j+... and λ j j 2s/N. Eigenvalues of the RFL are smaller than the ones of SFL: λ j λ s j for all j N. EIGENFUNCTIONS: φ j are the normalized eigenfunctions, are only Hölder continuous up to the boundary, namely φ j C s (Ω). (J. Serra - X. Ros-Oton) Lateral boundary conditions for the RFL u(t, x) = 0, in (0, ) ( R N \ Ω ). The Green function of RFL satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = s γ References. (K4) Bounds proven by Bogdan, Grzywny, Jakubowski, Kulczycki, Ryznar ( ). Eigenvalues: Blumental-Getoor (959), Chen-Song (2005)
24 Examples of operators L Definition via the hypersingular kernel in R N, restricted to functions that are zero outside Ω. The (Restricted) Fractional Laplacian operator (RFL) ( Ω ) s g(x) g(z) g(x) = c N,s P.V. dz, with supp(g) Ω. R N x z N+2s where s (0, ) and c N,s > 0 is a normalization constant. ( Ω ) s is a self-adjoint operator on L 2 (Ω) with a discrete spectrum: EIGENVALUES: 0 < λ λ 2... λ j λ j+... and λ j j 2s/N. Eigenvalues of the RFL are smaller than the ones of SFL: λ j λ s j for all j N. EIGENFUNCTIONS: φ j are the normalized eigenfunctions, are only Hölder continuous up to the boundary, namely φ j C s (Ω). (J. Serra - X. Ros-Oton) Lateral boundary conditions for the RFL u(t, x) = 0, in (0, ) ( R N \ Ω ). The Green function of RFL satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = s γ References. (K4) Bounds proven by Bogdan, Grzywny, Jakubowski, Kulczycki, Ryznar ( ). Eigenvalues: Blumental-Getoor (959), Chen-Song (2005)
25 Examples of operators L Introduced in 2003 by Bogdan, Burdzy and Chen. Censored Fractional Laplacians (CFL) a(x, y) Lf (x) = P.V. (f (x) f (y)) x y dy, with N+2s 2 < s <, Ω where a(x, y) is a measurable, symmetric function bounded between two positive constants, satisfying some further assumptions; for instance a C (Ω Ω). The Green function K(x, y) satisfies (K4), proven by Chen, Kim and Song (200) ( ) ( ) δ γ (x) δ γ K(x, y) x y N 2s x y (y) γ x y, with γ = s γ 2. Remarks. This is a third model of Dirichlet fractional Laplacian when [ a(x, y) = const ]. This is not equivalent to SFL nor to RFL. References. Roughly speaking, s (0, /2] corresponds to Neumann boundary conditions. K. Bogdan, K. Burdzy, K., Z.-Q. Chen. Censored stable processes. Probab. Theory Relat. Fields (2003) Z.-Q. Chen, P. Kim, R. Song, Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields (200)
26 Examples of operators L Introduced in 2003 by Bogdan, Burdzy and Chen. Censored Fractional Laplacians (CFL) a(x, y) Lf (x) = P.V. (f (x) f (y)) x y dy, with N+2s 2 < s <, Ω where a(x, y) is a measurable, symmetric function bounded between two positive constants, satisfying some further assumptions; for instance a C (Ω Ω). The Green function K(x, y) satisfies (K4), proven by Chen, Kim and Song (200) ( ) ( ) δ γ (x) δ γ K(x, y) x y N 2s x y (y) γ x y, with γ = s γ 2. Remarks. This is a third model of Dirichlet fractional Laplacian when [ a(x, y) = const ]. This is not equivalent to SFL nor to RFL. References. Roughly speaking, s (0, /2] corresponds to Neumann boundary conditions. K. Bogdan, K. Burdzy, K., Z.-Q. Chen. Censored stable processes. Probab. Theory Relat. Fields (2003) Z.-Q. Chen, P. Kim, R. Song, Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields (200)
27 Examples of operators L Introduced in 2003 by Bogdan, Burdzy and Chen. Censored Fractional Laplacians (CFL) a(x, y) Lf (x) = P.V. (f (x) f (y)) x y dy, with N+2s 2 < s <, Ω where a(x, y) is a measurable, symmetric function bounded between two positive constants, satisfying some further assumptions; for instance a C (Ω Ω). The Green function K(x, y) satisfies (K4), proven by Chen, Kim and Song (200) ( ) ( ) δ γ (x) δ γ K(x, y) x y N 2s x y (y) γ x y, with γ = s γ 2. Remarks. This is a third model of Dirichlet fractional Laplacian when [ a(x, y) = const ]. This is not equivalent to SFL nor to RFL. References. Roughly speaking, s (0, /2] corresponds to Neumann boundary conditions. K. Bogdan, K. Burdzy, K., Z.-Q. Chen. Censored stable processes. Probab. Theory Relat. Fields (2003) Z.-Q. Chen, P. Kim, R. Song, Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields (200)
28 About the kernels About the kernels of spectral nonlocal operators. Most of the examples of nonlocal operators, but the SFL, admit a representation with a kernel, namely ( ) Lf (x) = P.V. f (x) f (y) K(x, y) dy R N A natural question is: does the SFL admit such a representation? As far as we know, a description of such kernel was not known. Let A be a uniformly elliptic linear operator, we can define the s th power of A, Lg(x) = A s g(x) = Γ( s) The main questions are: 0 ( e ta g(x) g(x) ) dt t +s Does L admit a representation with a Kernel? If yes, What do we know about the Kernel? Positivity, support, (sharp) estimates? The key point is the knowledge of the Heat Kernel of A.
29 About the kernels About the kernels of spectral nonlocal operators. Most of the examples of nonlocal operators, but the SFL, admit a representation with a kernel, namely ( ) Lf (x) = P.V. f (x) f (y) K(x, y) dy R N A natural question is: does the SFL admit such a representation? As far as we know, a description of such kernel was not known. Let A be a uniformly elliptic linear operator, we can define the s th power of A, Lg(x) = A s g(x) = Γ( s) The main questions are: 0 ( e ta g(x) g(x) ) dt t +s Does L admit a representation with a Kernel? If yes, What do we know about the Kernel? Positivity, support, (sharp) estimates? The key point is the knowledge of the Heat Kernel of A.
30 About the kernels About the kernels of spectral nonlocal operators. Most of the examples of nonlocal operators, but the SFL, admit a representation with a kernel, namely ( ) Lf (x) = P.V. f (x) f (y) K(x, y) dy R N A natural question is: does the SFL admit such a representation? As far as we know, a description of such kernel was not known. Let A be a uniformly elliptic linear operator, we can define the s th power of A, Lg(x) = A s g(x) = Γ( s) The main questions are: 0 ( e ta g(x) g(x) ) dt t +s Does L admit a representation with a Kernel? If yes, What do we know about the Kernel? Positivity, support, (sharp) estimates? The key point is the knowledge of the Heat Kernel of A.
31 About the kernels About the kernels of spectral nonlocal operators. Most of the examples of nonlocal operators, but the SFL, admit a representation with a kernel, namely ( ) Lf (x) = P.V. f (x) f (y) K(x, y) dy R N A natural question is: does the SFL admit such a representation? As far as we know, a description of such kernel was not known. Let A be a uniformly elliptic linear operator, we can define the s th power of A, Lg(x) = A s g(x) = Γ( s) The main questions are: 0 ( e ta g(x) g(x) ) dt t +s Does L admit a representation with a Kernel? If yes, What do we know about the Kernel? Positivity, support, (sharp) estimates? The key point is the knowledge of the Heat Kernel of A.
32 About the kernels Lemma. Spectral kernels. (M.B. A. Figalli and J. L. Vázquez) Let s (0, ), and let L be the s th -power of a linear operator A, and Φ dist(, Ω) the first eigenfunction of A. Let H(t, x, y) be the Heat Kernel of A, and assume that it satisfies the following bounds: there exist constants c 0, c, c 2 > 0 such that H(t, x, y) [ ] [ ] Φ(x) Φ(y) e x y 2 c k t for all 0 < t. t γ/2 t γ/2 t N/2 (c k {c 0, c } maybe different in the upper and lower bounds) and also 0 H(t, x, y) c 2Φ (x)φ (y) for all t. Then the operator L admits a kernel representation, with ( ) ( ) Φ(x) Φ(y) K(x, y) x y N+2s x y γ x y γ In particular, the kernel K is supported in Ω Ω. One line (formal) proof: (recalling that when x y, we have δ x(y) = 0) K(x, y) = Lδ y(x) = Γ( s) (H(t, x, y) δ x(y)) 0 dt t +s = Γ( s) 0 H(t, x, y) t +s dt.
33 About the kernels Lemma. Spectral kernels. (M.B. A. Figalli and J. L. Vázquez) Let s (0, ), and let L be the s th -power of a linear operator A, and Φ dist(, Ω) the first eigenfunction of A. Let H(t, x, y) be the Heat Kernel of A, and assume that it satisfies the following bounds: there exist constants c 0, c, c 2 > 0 such that H(t, x, y) [ ] [ ] Φ(x) Φ(y) e x y 2 c k t for all 0 < t. t γ/2 t γ/2 t N/2 (c k {c 0, c } maybe different in the upper and lower bounds) and also 0 H(t, x, y) c 2Φ (x)φ (y) for all t. Then the operator L admits a kernel representation, with ( ) ( ) Φ(x) Φ(y) K(x, y) x y N+2s x y γ x y γ In particular, the kernel K is supported in Ω Ω. One line (formal) proof: (recalling that when x y, we have δ x(y) = 0) K(x, y) = Lδ y(x) = Γ( s) (H(t, x, y) δ x(y)) 0 dt t +s = Γ( s) 0 H(t, x, y) t +s dt.
34 More Examples Spectral powers of uniformly elliptic operators. Consider a linear operator A in divergence form, with uniformly elliptic bounded measurable coefficients: A = N i(a ij j), s-power of A is: Lf (x) := A s f (x) := λ s ˆf k kφ k(x) i,j= L = A s satisfies (K3) estimates with γ = (K3) c 0,Ωφ (x) φ (y) K(x, y) c,ω x y N 2s k= ( ) ( ) φ(x) φ(y) x y x y [General class of intrinsically ultra-contractive operators, Davies and Simon JFA 984]. Fractional operators with rough kernels. Integral operators of Levy-type K(x, y) Lf (x) = P.V. (f (x + y) f (y)) dy. R N x y N+2s where K is measurable, symmetric, bounded between two positive constants, and K(x, y) K(x, x) χ x y < c x y σ, with 0 < s < σ, for some positive c > 0. We can allow even more general kernels. The Green function satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = s γ
35 More Examples Spectral powers of uniformly elliptic operators. Consider a linear operator A in divergence form, with uniformly elliptic bounded measurable coefficients: A = N i(a ij j), s-power of A is: Lf (x) := A s f (x) := λ s ˆf k kφ k(x) i,j= L = A s satisfies (K3) estimates with γ = (K3) c 0,Ωφ (x) φ (y) K(x, y) c,ω x y N 2s k= ( ) ( ) φ(x) φ(y) x y x y [General class of intrinsically ultra-contractive operators, Davies and Simon JFA 984]. Fractional operators with rough kernels. Integral operators of Levy-type K(x, y) Lf (x) = P.V. (f (x + y) f (y)) dy. R N x y N+2s where K is measurable, symmetric, bounded between two positive constants, and K(x, y) K(x, x) χ x y < c x y σ, with 0 < s < σ, for some positive c > 0. We can allow even more general kernels. The Green function satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = s γ
36 More Examples Sums of two Restricted Fractional Laplacians. Operators of the form L = ( Ω ) s + ( Ω ) σ, with 0 < σ < s, where ( Ω ) s is the RFL. Satisfy (K4) with γ = s. Sum of the Laplacian and operators with general kernels. In the case L = a + A s, with 0 < s < and a 0, where ( ) A sf (x) = P.V. f (x + y) f (y) f (x) yχ y χ y dν(y), R N the measure ν on R N \ {0} is invariant under rotations around origin and satisfies x 2 dν(y) <, together with other assumptions. R N Relativistic stable processes. In the case ( s L = c c /s ), with c > 0, and 0 < s. The Green function K(x, y) of L satisfies assumption (K 4) with γ = s. Many other interesting examples. Schrödinger equations for non-symmetric diffusions, Gradient perturbation of RFL... References. The above mentioned bounds for the Green functions have been proven by Chen, Kim, Song and Vondracek (2007, 200, 202, 203).
37 More Examples Sums of two Restricted Fractional Laplacians. Operators of the form L = ( Ω ) s + ( Ω ) σ, with 0 < σ < s, where ( Ω ) s is the RFL. Satisfy (K4) with γ = s. Sum of the Laplacian and operators with general kernels. In the case L = a + A s, with 0 < s < and a 0, where ( ) A sf (x) = P.V. f (x + y) f (y) f (x) yχ y χ y dν(y), R N the measure ν on R N \ {0} is invariant under rotations around origin and satisfies x 2 dν(y) <, together with other assumptions. R N Relativistic stable processes. In the case ( s L = c c /s ), with c > 0, and 0 < s. The Green function K(x, y) of L satisfies assumption (K 4) with γ = s. Many other interesting examples. Schrödinger equations for non-symmetric diffusions, Gradient perturbation of RFL... References. The above mentioned bounds for the Green functions have been proven by Chen, Kim, Song and Vondracek (2007, 200, 202, 203).
38 More Examples Sums of two Restricted Fractional Laplacians. Operators of the form L = ( Ω ) s + ( Ω ) σ, with 0 < σ < s, where ( Ω ) s is the RFL. Satisfy (K4) with γ = s. Sum of the Laplacian and operators with general kernels. In the case L = a + A s, with 0 < s < and a 0, where ( ) A sf (x) = P.V. f (x + y) f (y) f (x) yχ y χ y dν(y), R N the measure ν on R N \ {0} is invariant under rotations around origin and satisfies x 2 dν(y) <, together with other assumptions. R N Relativistic stable processes. In the case ( s L = c c /s ), with c > 0, and 0 < s. The Green function K(x, y) of L satisfies assumption (K 4) with γ = s. Many other interesting examples. Schrödinger equations for non-symmetric diffusions, Gradient perturbation of RFL... References. The above mentioned bounds for the Green functions have been proven by Chen, Kim, Song and Vondracek (2007, 200, 202, 203).
39 More Examples Sums of two Restricted Fractional Laplacians. Operators of the form L = ( Ω ) s + ( Ω ) σ, with 0 < σ < s, where ( Ω ) s is the RFL. Satisfy (K4) with γ = s. Sum of the Laplacian and operators with general kernels. In the case L = a + A s, with 0 < s < and a 0, where ( ) A sf (x) = P.V. f (x + y) f (y) f (x) yχ y χ y dν(y), R N the measure ν on R N \ {0} is invariant under rotations around origin and satisfies x 2 dν(y) <, together with other assumptions. R N Relativistic stable processes. In the case ( s L = c c /s ), with c > 0, and 0 < s. The Green function K(x, y) of L satisfies assumption (K 4) with γ = s. Many other interesting examples. Schrödinger equations for non-symmetric diffusions, Gradient perturbation of RFL... References. The above mentioned bounds for the Green functions have been proven by Chen, Kim, Song and Vondracek (2007, 200, 202, 203).
40 Basic Theory The Dual problem Existence and uniqueness First set of estimates For the rest of the talk we deal with the special case: F(u) = u m := u m u
41 The dual formulation of the problem The dual formulation of the problem. Recall the homogeneous Cauchy-Dirichlet problem: t u = L u m, in (0, + ) Ω (CDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. We can formulate a dual problem, using the inverse L as follows where t U = u m, U(t, x) := L [u(t, )](x) = Ω u(t, y)k(x, y) dy. This formulation encodes all the possible lateral boundary conditions in the inverse operator L. Remark. This formulation has been used before by Pierre, Vázquez [...] to prove (in the R N case) uniqueness of the fundamental solution, i.e. the solution corresponding to u 0 = δ x0, known as the Barenblatt solution.
42 The dual formulation of the problem The dual formulation of the problem. Recall the homogeneous Cauchy-Dirichlet problem: t u = L u m, in (0, + ) Ω (CDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. We can formulate a dual problem, using the inverse L as follows where t U = u m, U(t, x) := L [u(t, )](x) = Ω u(t, y)k(x, y) dy. This formulation encodes all the possible lateral boundary conditions in the inverse operator L. Remark. This formulation has been used before by Pierre, Vázquez [...] to prove (in the R N case) uniqueness of the fundamental solution, i.e. the solution corresponding to u 0 = δ x0, known as the Barenblatt solution.
43 Existence and uniqueness of weak dual solutions Recall that Φ dist(, Ω) γ and w L Φ (Ω) = Ω w(x)φ (x) dx. Weak Dual Solutions for the Cauchy Dirichlet Problem (CDP) A function u is a weak dual solution to the Cauchy-Dirichlet problem (CDP) for the equation tu + Lu m = 0 in Q T = (0, T) Ω if: u C((0, T) : L Φ (Ω)), u m L ( (0, T) : L Φ (Ω) ) ; The following identity holds for every ψ/φ C c((0, T) : L (Ω)) : T 0 Ω L (u) ψ t dx dt T u C([0, T) : L Φ (Ω)) and u(0, x) = u 0 L Φ (Ω). 0 Ω u m ψ dx dt = 0. Theorem. Existence and Uniqueness (M.B. and J. L. Vázquez) For every nonnegative u 0 L Φ (Ω) there exists a unique minimal weak dual solution to the (CDP). Such a solution is obtained as the monotone limit of the semigroup (mild) solutions that exist and are unique. The minimal weak dual solution is continuous in the weighted space u C([0, ) : L Φ (Ω)). In this class of solutions the standard comparison result holds. Remarks. Mild solutions (by Crandall and Pierre) are weak dual solutions. Weak dual solutions are very weak solutions.
44 Existence and uniqueness of weak dual solutions Recall that Φ dist(, Ω) γ and w L Φ (Ω) = Ω w(x)φ (x) dx. Weak Dual Solutions for the Cauchy Dirichlet Problem (CDP) A function u is a weak dual solution to the Cauchy-Dirichlet problem (CDP) for the equation tu + Lu m = 0 in Q T = (0, T) Ω if: u C((0, T) : L Φ (Ω)), u m L ( (0, T) : L Φ (Ω) ) ; The following identity holds for every ψ/φ C c((0, T) : L (Ω)) : T 0 Ω L (u) ψ t dx dt T u C([0, T) : L Φ (Ω)) and u(0, x) = u 0 L Φ (Ω). 0 Ω u m ψ dx dt = 0. Theorem. Existence and Uniqueness (M.B. and J. L. Vázquez) For every nonnegative u 0 L Φ (Ω) there exists a unique minimal weak dual solution to the (CDP). Such a solution is obtained as the monotone limit of the semigroup (mild) solutions that exist and are unique. The minimal weak dual solution is continuous in the weighted space u C([0, ) : L Φ (Ω)). In this class of solutions the standard comparison result holds. Remarks. Mild solutions (by Crandall and Pierre) are weak dual solutions. Weak dual solutions are very weak solutions.
45 Existence and uniqueness of weak dual solutions Recall that Φ dist(, Ω) γ and w L Φ (Ω) = Ω w(x)φ (x) dx. Weak Dual Solutions for the Cauchy Dirichlet Problem (CDP) A function u is a weak dual solution to the Cauchy-Dirichlet problem (CDP) for the equation tu + Lu m = 0 in Q T = (0, T) Ω if: u C((0, T) : L Φ (Ω)), u m L ( (0, T) : L Φ (Ω) ) ; The following identity holds for every ψ/φ C c((0, T) : L (Ω)) : T 0 Ω L (u) ψ t dx dt T u C([0, T) : L Φ (Ω)) and u(0, x) = u 0 L Φ (Ω). 0 Ω u m ψ dx dt = 0. Theorem. Existence and Uniqueness (M.B. and J. L. Vázquez) For every nonnegative u 0 L Φ (Ω) there exists a unique minimal weak dual solution to the (CDP). Such a solution is obtained as the monotone limit of the semigroup (mild) solutions that exist and are unique. The minimal weak dual solution is continuous in the weighted space u C([0, ) : L Φ (Ω)). In this class of solutions the standard comparison result holds. Remarks. Mild solutions (by Crandall and Pierre) are weak dual solutions. Weak dual solutions are very weak solutions.
46 Existence and uniqueness of weak dual solutions Recall that Φ dist(, Ω) γ and w L Φ (Ω) = Ω w(x)φ (x) dx. Weak Dual Solutions for the Cauchy Dirichlet Problem (CDP) A function u is a weak dual solution to the Cauchy-Dirichlet problem (CDP) for the equation tu + Lu m = 0 in Q T = (0, T) Ω if: u C((0, T) : L Φ (Ω)), u m L ( (0, T) : L Φ (Ω) ) ; The following identity holds for every ψ/φ C c((0, T) : L (Ω)) : T 0 Ω L (u) ψ t dx dt T u C([0, T) : L Φ (Ω)) and u(0, x) = u 0 L Φ (Ω). 0 Ω u m ψ dx dt = 0. Theorem. Existence and Uniqueness (M.B. and J. L. Vázquez) For every nonnegative u 0 L Φ (Ω) there exists a unique minimal weak dual solution to the (CDP). Such a solution is obtained as the monotone limit of the semigroup (mild) solutions that exist and are unique. The minimal weak dual solution is continuous in the weighted space u C([0, ) : L Φ (Ω)). In this class of solutions the standard comparison result holds. Remarks. Mild solutions (by Crandall and Pierre) are weak dual solutions. Weak dual solutions are very weak solutions.
47 Basic Estimates Theorem. First Pointwise Estimates. Let u 0 be a nonnegative weak dual solution to Problem (CDP). Then, for almost every 0 t 0 t and almost every x 0 Ω, we have ( t0 t ) m m u m (t 0, x 0) Theorem. (Absolute upper bounds) Ω u(t 0, x) u(t, x) t t 0 K(x, x 0) dx (M.B. and J. L. Vázquez) ( t t 0 ) m m u m (t, x 0). (M.B. & J. L. Vázquez) Let u be a weak dual solution, then there exists a constant κ 0 > 0 depending only on N, s, m, Ω (but not on u 0!!), such that under the minimal assumption (K): u(t) L (Ω) t κ0 m, for all t > 0. Theorem. (Smoothing effects) (M.B. & J. L. Vázquez) Let ϑ γ = /[2s + (N + γ)(m )] and assume (K2). There exists κ > 0 such that: u(t) L (Ω) κ u(t) 2sϑγ Nϑγ t L Φ (Ω) κ u0 2sϑγ Nϑγ t L Φ (Ω) for all t > 0. Assuming only (K), the above bound holds with L and ϑ 0, instead of L Φ and ϑ γ. More details about this first part online:
48 Basic Estimates Theorem. First Pointwise Estimates. Let u 0 be a nonnegative weak dual solution to Problem (CDP). Then, for almost every 0 t 0 t and almost every x 0 Ω, we have ( t0 t ) m m u m (t 0, x 0) Theorem. (Absolute upper bounds) Ω u(t 0, x) u(t, x) t t 0 K(x, x 0) dx (M.B. and J. L. Vázquez) ( t t 0 ) m m u m (t, x 0). (M.B. & J. L. Vázquez) Let u be a weak dual solution, then there exists a constant κ 0 > 0 depending only on N, s, m, Ω (but not on u 0!!), such that under the minimal assumption (K): u(t) L (Ω) t κ0 m, for all t > 0. Theorem. (Smoothing effects) (M.B. & J. L. Vázquez) Let ϑ γ = /[2s + (N + γ)(m )] and assume (K2). There exists κ > 0 such that: u(t) L (Ω) κ u(t) 2sϑγ Nϑγ t L Φ (Ω) κ u0 2sϑγ Nϑγ t L Φ (Ω) for all t > 0. Assuming only (K), the above bound holds with L and ϑ 0, instead of L Φ and ϑ γ. More details about this first part online:
49 Basic Estimates Theorem. First Pointwise Estimates. Let u 0 be a nonnegative weak dual solution to Problem (CDP). Then, for almost every 0 t 0 t and almost every x 0 Ω, we have ( t0 t ) m m u m (t 0, x 0) Theorem. (Absolute upper bounds) Ω u(t 0, x) u(t, x) t t 0 K(x, x 0) dx (M.B. and J. L. Vázquez) ( t t 0 ) m m u m (t, x 0). (M.B. & J. L. Vázquez) Let u be a weak dual solution, then there exists a constant κ 0 > 0 depending only on N, s, m, Ω (but not on u 0!!), such that under the minimal assumption (K): u(t) L (Ω) t κ0 m, for all t > 0. Theorem. (Smoothing effects) (M.B. & J. L. Vázquez) Let ϑ γ = /[2s + (N + γ)(m )] and assume (K2). There exists κ > 0 such that: u(t) L (Ω) κ u(t) 2sϑγ Nϑγ t L Φ (Ω) κ u0 2sϑγ Nϑγ t L Φ (Ω) for all t > 0. Assuming only (K), the above bound holds with L and ϑ 0, instead of L Φ and ϑ γ. More details about this first part online:
50 Basic Estimates Theorem. First Pointwise Estimates. Let u 0 be a nonnegative weak dual solution to Problem (CDP). Then, for almost every 0 t 0 t and almost every x 0 Ω, we have ( t0 t ) m m u m (t 0, x 0) Theorem. (Absolute upper bounds) Ω u(t 0, x) u(t, x) t t 0 K(x, x 0) dx (M.B. and J. L. Vázquez) ( t t 0 ) m m u m (t, x 0). (M.B. & J. L. Vázquez) Let u be a weak dual solution, then there exists a constant κ 0 > 0 depending only on N, s, m, Ω (but not on u 0!!), such that under the minimal assumption (K): u(t) L (Ω) t κ0 m, for all t > 0. Theorem. (Smoothing effects) (M.B. & J. L. Vázquez) Let ϑ γ = /[2s + (N + γ)(m )] and assume (K2). There exists κ > 0 such that: u(t) L (Ω) κ u(t) 2sϑγ Nϑγ t L Φ (Ω) κ u0 2sϑγ Nϑγ t L Φ (Ω) for all t > 0. Assuming only (K), the above bound holds with L and ϑ 0, instead of L Φ and ϑ γ. More details about this first part online:
51 Sharp Boundary Behaviour Upper Boundary Estimates Infinite Speed of Propagation Lower Boundary Estimates
52 Upper boundary estimates Sharp Upper boundary estimates
53 Upper boundary estimates Theorem. (Upper boundary behaviour) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold. Let u 0 be a weak dual solution to the (CDP). Let σ (0, ] be σ = 2sm γ(m ) Then, there exists a computable constant κ > 0, depending only on N, s, m, and Ω, (but not on u 0!!) such that for all t 0 and all x Ω : u(t, x) κ Φ (x) σ m t m σγ dist(x, Ω) m t m When σ = we have sharp boundary estimates: we will show lower bounds with matching powers. When σ < the estimates are not sharp in all cases: The solution by separation of variables U(t, x) = S(x)t /(m ) (asymptotic behaviour) behaves like Φ σ/m t /(m ). We will show that for small data, the boundary behaviour is different. In examples, σ < only happens for SFL-type, where γ =, and s can be small, 0 < s < /2 /(2m).
54 Upper boundary estimates Theorem. (Upper boundary behaviour) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold. Let u 0 be a weak dual solution to the (CDP). Let σ (0, ] be σ = 2sm γ(m ) Then, there exists a computable constant κ > 0, depending only on N, s, m, and Ω, (but not on u 0!!) such that for all t 0 and all x Ω : u(t, x) κ Φ (x) σ m t m σγ dist(x, Ω) m t m When σ = we have sharp boundary estimates: we will show lower bounds with matching powers. When σ < the estimates are not sharp in all cases: The solution by separation of variables U(t, x) = S(x)t /(m ) (asymptotic behaviour) behaves like Φ σ/m t /(m ). We will show that for small data, the boundary behaviour is different. In examples, σ < only happens for SFL-type, where γ =, and s can be small, 0 < s < /2 /(2m).
55 Upper boundary estimates Theorem. (Upper boundary behaviour) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold. Let u 0 be a weak dual solution to the (CDP). Let σ (0, ] be σ = 2sm γ(m ) Then, there exists a computable constant κ > 0, depending only on N, s, m, and Ω, (but not on u 0!!) such that for all t 0 and all x Ω : u(t, x) κ Φ (x) σ m t m σγ dist(x, Ω) m t m When σ = we have sharp boundary estimates: we will show lower bounds with matching powers. When σ < the estimates are not sharp in all cases: The solution by separation of variables U(t, x) = S(x)t /(m ) (asymptotic behaviour) behaves like Φ σ/m t /(m ). We will show that for small data, the boundary behaviour is different. In examples, σ < only happens for SFL-type, where γ =, and s can be small, 0 < s < /2 /(2m).
56 Upper boundary estimates Theorem. (Upper boundary behaviour) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold. Let u 0 be a weak dual solution to the (CDP). Let σ (0, ] be σ = 2sm γ(m ) Then, there exists a computable constant κ > 0, depending only on N, s, m, and Ω, (but not on u 0!!) such that for all t 0 and all x Ω : u(t, x) κ Φ (x) σ m t m σγ dist(x, Ω) m t m When σ = we have sharp boundary estimates: we will show lower bounds with matching powers. When σ < the estimates are not sharp in all cases: The solution by separation of variables U(t, x) = S(x)t /(m ) (asymptotic behaviour) behaves like Φ σ/m t /(m ). We will show that for small data, the boundary behaviour is different. In examples, σ < only happens for SFL-type, where γ =, and s can be small, 0 < s < /2 /(2m).
57 Upper boundary estimates Now consider σ < which means taking s small respect to γ, as follows σ = 2sm γ(m ) < i.e. 0 < s < γ 2 γ 2m. The previous upper bounds are not sharp for small times for small data : Theorem. (Upper boundary behaviour II) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold, and let σ <. Let u 0 be a weak dual solution to (CDP). There exists a constant κ 0 > 0 that depends only on N, s, m, λ, and Ω, such that the following holds: given T > 0, assume that for all x Ω 0 u 0 (x) ε T Φ (x) with ε T = κ 0 e λt. Then for all (t, x) (0, T] Ω we have u(t, x) κ 0 Φ (x). This phenomenon does not happen in the linear case, for u t + Lu = 0. It does not happen either in the nonlinear elliptic case LS m = λs, where S Φ σ/m. This theorem applies for instance to spectral power of an elliptic operators, L = A s. Recall that the domain Ω is smooth, at least C,
58 Upper boundary estimates Now consider σ < which means taking s small respect to γ, as follows σ = 2sm γ(m ) < i.e. 0 < s < γ 2 γ 2m. The previous upper bounds are not sharp for small times for small data : Theorem. (Upper boundary behaviour II) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold, and let σ <. Let u 0 be a weak dual solution to (CDP). There exists a constant κ 0 > 0 that depends only on N, s, m, λ, and Ω, such that the following holds: given T > 0, assume that for all x Ω 0 u 0 (x) ε T Φ (x) with ε T = κ 0 e λt. Then for all (t, x) (0, T] Ω we have u(t, x) κ 0 Φ (x). This phenomenon does not happen in the linear case, for u t + Lu = 0. It does not happen either in the nonlinear elliptic case LS m = λs, where S Φ σ/m. This theorem applies for instance to spectral power of an elliptic operators, L = A s. Recall that the domain Ω is smooth, at least C,
59 Upper boundary estimates Now consider σ < which means taking s small respect to γ, as follows σ = 2sm γ(m ) < i.e. 0 < s < γ 2 γ 2m. The previous upper bounds are not sharp for small times for small data : Theorem. (Upper boundary behaviour II) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold, and let σ <. Let u 0 be a weak dual solution to (CDP). There exists a constant κ 0 > 0 that depends only on N, s, m, λ, and Ω, such that the following holds: given T > 0, assume that for all x Ω 0 u 0 (x) ε T Φ (x) with ε T = κ 0 e λt. Then for all (t, x) (0, T] Ω we have u(t, x) κ 0 Φ (x). This phenomenon does not happen in the linear case, for u t + Lu = 0. It does not happen either in the nonlinear elliptic case LS m = λs, where S Φ σ/m. This theorem applies for instance to spectral power of an elliptic operators, L = A s. Recall that the domain Ω is smooth, at least C,
60 Upper boundary estimates Now consider σ < which means taking s small respect to γ, as follows σ = 2sm γ(m ) < i.e. 0 < s < γ 2 γ 2m. The previous upper bounds are not sharp for small times for small data : Theorem. (Upper boundary behaviour II) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold, and let σ <. Let u 0 be a weak dual solution to (CDP). There exists a constant κ 0 > 0 that depends only on N, s, m, λ, and Ω, such that the following holds: given T > 0, assume that for all x Ω 0 u 0 (x) ε T Φ (x) with ε T = κ 0 e λt. Then for all (t, x) (0, T] Ω we have u(t, x) κ 0 Φ (x). This phenomenon does not happen in the linear case, for u t + Lu = 0. It does not happen either in the nonlinear elliptic case LS m = λs, where S Φ σ/m. This theorem applies for instance to spectral power of an elliptic operators, L = A s. Recall that the domain Ω is smooth, at least C,
61 Infinite Speed of Propagation Infinite Speed of Propagation and Universal Lower Bounds
62 Infinite Speed of Propagation Theorem. (Universal lower bounds) (M.B., A. Figalli and J. L. Vázquez) Let L satisfy (A) and (A2), and assume that ( ) Lw(x) = P.V. w(x) w(y) K(x, y) dy, with K(x, y) c0 Φ (x)φ (y) x, y Ω. R N Let u 0 be a weak dual solution to the (CDP) corresponding to u 0 L Φ (Ω). Then there exists a constant κ 0 > 0, so that the following inequality holds: ( u(t, x) κ 0 t ) m m Φ (x) for all t > 0 and all x Ω. t t m Here t = κ u 0 (m ) and κ L Φ (Ω) 0, κ depend only on N, s, γ, m, c0, and Ω. Note that, for t t, the dependence on the initial data disappears u(t) κ 0 Φ t m t t. The assumption on the kernel K of L holds for all examples and represent somehow the worst case scenario for lower estimates. In many cases (RFL, CFL), K satisfies a stronger property: K κ Ω > 0 in Ω Ω. It is remarkable that this minimal behavior is sharp in some situations: for σ <, small data and for small times.
63 Infinite Speed of Propagation Theorem. (Universal lower bounds) (M.B., A. Figalli and J. L. Vázquez) Let L satisfy (A) and (A2), and assume that ( ) Lw(x) = P.V. w(x) w(y) K(x, y) dy, with K(x, y) c0 Φ (x)φ (y) x, y Ω. R N Let u 0 be a weak dual solution to the (CDP) corresponding to u 0 L Φ (Ω). Then there exists a constant κ 0 > 0, so that the following inequality holds: ( u(t, x) κ 0 t ) m m Φ (x) for all t > 0 and all x Ω. t t m Here t = κ u 0 (m ) and κ L Φ (Ω) 0, κ depend only on N, s, γ, m, c0, and Ω. Note that, for t t, the dependence on the initial data disappears u(t) κ 0 Φ t m t t. The assumption on the kernel K of L holds for all examples and represent somehow the worst case scenario for lower estimates. In many cases (RFL, CFL), K satisfies a stronger property: K κ Ω > 0 in Ω Ω. It is remarkable that this minimal behavior is sharp in some situations: for σ <, small data and for small times.
64 Infinite Speed of Propagation Theorem. (Universal lower bounds) (M.B., A. Figalli and J. L. Vázquez) Let L satisfy (A) and (A2), and assume that ( ) Lw(x) = P.V. w(x) w(y) K(x, y) dy, with K(x, y) c0 Φ (x)φ (y) x, y Ω. R N Let u 0 be a weak dual solution to the (CDP) corresponding to u 0 L Φ (Ω). Then there exists a constant κ 0 > 0, so that the following inequality holds: ( u(t, x) κ 0 t ) m m Φ (x) for all t > 0 and all x Ω. t t m Here t = κ u 0 (m ) and κ L Φ (Ω) 0, κ depend only on N, s, γ, m, c0, and Ω. Note that, for t t, the dependence on the initial data disappears u(t) κ 0 Φ t m t t. The assumption on the kernel K of L holds for all examples and represent somehow the worst case scenario for lower estimates. In many cases (RFL, CFL), K satisfies a stronger property: K κ Ω > 0 in Ω Ω. It is remarkable that this minimal behavior is sharp in some situations: for σ <, small data and for small times.
65 Infinite Speed of Propagation Theorem. (Universal lower bounds) (M.B., A. Figalli and J. L. Vázquez) Let L satisfy (A) and (A2), and assume that ( ) Lw(x) = P.V. w(x) w(y) K(x, y) dy, with K(x, y) c0 Φ (x)φ (y) x, y Ω. R N Let u 0 be a weak dual solution to the (CDP) corresponding to u 0 L Φ (Ω). Then there exists a constant κ 0 > 0, so that the following inequality holds: ( u(t, x) κ 0 t ) m m Φ (x) for all t > 0 and all x Ω. t t m Here t = κ u 0 (m ) and κ L Φ (Ω) 0, κ depend only on N, s, γ, m, c0, and Ω. Note that, for t t, the dependence on the initial data disappears u(t) κ 0 Φ t m t t. The assumption on the kernel K of L holds for all examples and represent somehow the worst case scenario for lower estimates. In many cases (RFL, CFL), K satisfies a stronger property: K κ Ω > 0 in Ω Ω. It is remarkable that this minimal behavior is sharp in some situations: for σ <, small data and for small times.
Integro-differential equations: Regularity theory and Pohozaev identities
Integro-differential equations: Regularity theory and Pohozaev identities Xavier Ros Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya PhD Thesis Advisor: Xavier Cabré Xavier
More informationAsymptotic behavior of the degenerate p Laplacian equation on bounded domains
Asymptotic behavior of the degenerate p Laplacian equation on bounded domains Diana Stan Instituto de Ciencias Matematicas (CSIC), Madrid, Spain UAM, September 19, 2011 Diana Stan (ICMAT & UAM) Nonlinear
More informationOn some weighted fractional porous media equations
On some weighted fractional porous media equations Gabriele Grillo Politecnico di Milano September 16 th, 2015 Anacapri Joint works with M. Muratori and F. Punzo Gabriele Grillo Weighted Fractional PME
More informationFree boundaries in fractional filtration equations
Free boundaries in fractional filtration equations Fernando Quirós Universidad Autónoma de Madrid Joint work with Arturo de Pablo, Ana Rodríguez and Juan Luis Vázquez International Conference on Free Boundary
More informationarxiv: v1 [math.ap] 18 May 2017
Littlewood-Paley-Stein functions for Schrödinger operators arxiv:175.6794v1 [math.ap] 18 May 217 El Maati Ouhabaz Dedicated to the memory of Abdelghani Bellouquid (2/2/1966 8/31/215) Abstract We study
More informationRecent result on porous medium equations with nonlocal pressure
Recent result on porous medium equations with nonlocal pressure Diana Stan Basque Center of Applied Mathematics joint work with Félix del Teso and Juan Luis Vázquez November 2016 4 th workshop on Fractional
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationNONLOCAL DIFFUSION EQUATIONS
NONLOCAL DIFFUSION EQUATIONS JULIO D. ROSSI (ALICANTE, SPAIN AND BUENOS AIRES, ARGENTINA) jrossi@dm.uba.ar http://mate.dm.uba.ar/ jrossi 2011 Non-local diffusion. The function J. Let J : R N R, nonnegative,
More informationHardy inequalities, heat kernels and wave propagation
Outline Hardy inequalities, heat kernels and wave propagation Basque Center for Applied Mathematics (BCAM) Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ Third Brazilian
More informationPROBLEMS IN UNBOUNDED CYLINDRICAL DOMAINS
PROBLEMS IN UNBOUNDED CYLINDRICAL DOMAINS PATRICK GUIDOTTI Mathematics Department, University of California, Patrick Guidotti, 103 Multipurpose Science and Technology Bldg, Irvine, CA 92697, USA 1. Introduction
More informationTransformations of Self-Similar Solutions for porous medium equations of fractional type
Transformations of Self-Similar Solutions for porous medium equations of fractional type Diana Stan, Félix del Teso and Juan Luis Vázquez arxiv:140.6877v1 [math.ap] 7 Feb 014 February 8, 014 Abstract We
More informationFractional order operators on bounded domains
on bounded domains Gerd Grubb Copenhagen University Geometry seminar, Stanford University September 23, 2015 1. Fractional-order pseudodifferential operators A prominent example of a fractional-order pseudodifferential
More informationHeat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control
Outline Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control IMDEA-Matemáticas & Universidad Autónoma de Madrid Spain enrique.zuazua@uam.es Analysis and control
More informationNonlinear aspects of Calderón-Zygmund theory
Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with
More informationThe role of Wolff potentials in the analysis of degenerate parabolic equations
The role of Wolff potentials in the analysis of degenerate parabolic equations September 19, 2011 Universidad Autonoma de Madrid Some elliptic background Part 1: Ellipticity The classical potential estimates
More informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationarxiv: v1 [math.ap] 25 Jul 2012
THE DIRICHLET PROBLEM FOR THE FRACTIONAL LAPLACIAN: REGULARITY UP TO THE BOUNDARY XAVIER ROS-OTON AND JOAQUIM SERRA arxiv:1207.5985v1 [math.ap] 25 Jul 2012 Abstract. We study the regularity up to the boundary
More informationTRANSPORT IN POROUS MEDIA
1 TRANSPORT IN POROUS MEDIA G. ALLAIRE CMAP, Ecole Polytechnique 1. Introduction 2. Main result in an unbounded domain 3. Asymptotic expansions with drift 4. Two-scale convergence with drift 5. The case
More informationLecture No 2 Degenerate Diffusion Free boundary problems
Lecture No 2 Degenerate Diffusion Free boundary problems Columbia University IAS summer program June, 2009 Outline We will discuss non-linear parabolic equations of slow diffusion. Our model is the porous
More informationGLOBAL ATTRACTIVITY IN A CLASS OF NONMONOTONE REACTION-DIFFUSION EQUATIONS WITH TIME DELAY
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 2009 GLOBAL ATTRACTIVITY IN A CLASS OF NONMONOTONE REACTION-DIFFUSION EQUATIONS WITH TIME DELAY XIAO-QIANG ZHAO ABSTRACT. The global attractivity
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationHOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS. 1. Introduction
HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS CARMEN CORTAZAR, MANUEL ELGUETA, ULIO D. ROSSI, AND NOEMI WOLANSKI Abstract. We present a model for
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationEvolution problems involving the fractional Laplace operator: HUM control and Fourier analysis
Evolution problems involving the fractional Laplace operator: HUM control and Fourier analysis Umberto Biccari joint work with Enrique Zuazua BCAM - Basque Center for Applied Mathematics NUMERIWAVES group
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationBOUNDARY REGULARITY, POHOZAEV IDENTITIES AND NONEXISTENCE RESULTS
BOUNDARY REGULARITY, POHOZAEV IDENTITIES AND NONEXISTENCE RESULTS XAVIER ROS-OTON Abstract. In this expository paper we survey some recent results on Dirichlet problems of the form Lu = f(x, u) in, u 0
More informationLarge Solutions for Fractional Laplacian Operators
Some results of the PhD thesis Nicola Abatangelo Advisors: Louis Dupaigne, Enrico Valdinoci Université Libre de Bruxelles October 16 th, 2015 The operator Fix s (0, 1) and consider a measurable function
More informationON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES
ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES RODRIGO BAÑUELOS, TADEUSZ KULCZYCKI, AND PEDRO J. MÉNDEZ-HERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric
More informationEventual Positivity of Operator Semigroups
Eventual Positivity of Operator Semigroups Jochen Glück Ulm University Positivity IX, 17 May 21 May 2017 Joint work with Daniel Daners (University of Sydney) and James B. Kennedy (University of Lisbon)
More informationThe Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge
The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij
More informationElliptic Operators with Unbounded Coefficients
Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli Motivation Consider the Stochastic Differential
More informationBlow-up for a Nonlocal Nonlinear Diffusion Equation with Source
Revista Colombiana de Matemáticas Volumen 46(2121, páginas 1-13 Blow-up for a Nonlocal Nonlinear Diffusion Equation with Source Explosión para una ecuación no lineal de difusión no local con fuente Mauricio
More informationCOMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL. Ross G. Pinsky
COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL Ross G. Pinsky Department of Mathematics Technion-Israel Institute of Technology Haifa, 32000 Israel
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationComm. Nonlin. Sci. and Numer. Simul., 12, (2007),
Comm. Nonlin. Sci. and Numer. Simul., 12, (2007), 1390-1394. 1 A Schrödinger singular perturbation problem A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu
More informationOPTIMAL REGULARITY FOR A TWO-PHASE FREE BOUNDARY PROBLEM RULED BY THE INFINITY LAPLACIAN DAMIÃO J. ARAÚJO, EDUARDO V. TEIXEIRA AND JOSÉ MIGUEL URBANO
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 18 55 OPTIMAL REGULARITY FOR A TWO-PHASE FREE BOUNDARY PROBLEM RULED BY THE INFINITY LAPLACIAN DAMIÃO J. ARAÚJO, EDUARDO
More informationParameter Dependent Quasi-Linear Parabolic Equations
CADERNOS DE MATEMÁTICA 4, 39 33 October (23) ARTIGO NÚMERO SMA#79 Parameter Dependent Quasi-Linear Parabolic Equations Cláudia Buttarello Gentile Departamento de Matemática, Universidade Federal de São
More informationFractional Cahn-Hilliard equation
Fractional Cahn-Hilliard equation Goro Akagi Mathematical Institute, Tohoku University Abstract In this note, we review a recent joint work with G. Schimperna and A. Segatti (University of Pavia, Italy)
More informationAn Operator Theoretical Approach to Nonlocal Differential Equations
An Operator Theoretical Approach to Nonlocal Differential Equations Joshua Lee Padgett Department of Mathematics and Statistics Texas Tech University Analysis Seminar November 27, 2017 Joshua Lee Padgett
More informationTOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017
TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationA Mean Field Equation as Limit of Nonlinear Diffusions with Fractional Laplacian Operators
A Mean Field Equation as Limit of Nonlinear Diffusions with Fractional Laplacian Operators Sylvia Serfaty and Juan Luis Vázquez June 2012 Abstract In the limit of a nonlinear diffusion model involving
More informationarxiv:math/ v2 [math.ap] 3 Oct 2006
THE TAYLOR SERIES OF THE GAUSSIAN KERNEL arxiv:math/0606035v2 [math.ap] 3 Oct 2006 L. ESCAURIAZA From some people one can learn more than mathematics Abstract. We describe a formula for the Taylor series
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationTotal Variation Flow and Sign Fast Diffusion in one dimension
Total Variation Flow and Sign Fast Diffusion in one dimension Matteo Bonforte a and Alessio Figalli b August 17, 2011 Abstract We consider the dynamics of the Total Variation Flow (TVF) u t = div(du/ Du
More informationRegularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133
More informationVelocity averaging a general framework
Outline Velocity averaging a general framework Martin Lazar BCAM ERC-NUMERIWAVES Seminar May 15, 2013 Joint work with D. Mitrović, University of Montenegro, Montenegro Outline Outline 1 2 L p, p >= 2 setting
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationSwitching, sparse and averaged control
Switching, sparse and averaged control Enrique Zuazua Ikerbasque & BCAM Basque Center for Applied Mathematics Bilbao - Basque Country- Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ WG-BCAM, February
More informationOn Schrödinger equations with inverse-square singular potentials
On Schrödinger equations with inverse-square singular potentials Veronica Felli Dipartimento di Statistica University of Milano Bicocca veronica.felli@unimib.it joint work with Elsa M. Marchini and Susanna
More informationIntroduction to Aspects of Multiscale Modeling as Applied to Porous Media
Introduction to Aspects of Multiscale Modeling as Applied to Porous Media Part III Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and
More informationObstacle Problems Involving The Fractional Laplacian
Obstacle Problems Involving The Fractional Laplacian Donatella Danielli and Sandro Salsa January 27, 2017 1 Introduction Obstacle problems involving a fractional power of the Laplace operator appear in
More informationInégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur.
Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur. Luc Miller Université Paris Ouest Nanterre La Défense, France Pde s, Dispersion, Scattering
More informationOn non negative solutions of some quasilinear elliptic inequalities
On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationPartial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation:
Chapter 7 Heat Equation Partial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation: u t = ku x x, x, t > (7.1) Here k is a constant
More informationPerturbations of singular solutions to Gelfand s problem p.1
Perturbations of singular solutions to Gelfand s problem Juan Dávila (Universidad de Chile) In celebration of the 60th birthday of Ireneo Peral February 2007 collaboration with Louis Dupaigne (Université
More informationP(E t, Ω)dt, (2) 4t has an advantage with respect. to the compactly supported mollifiers, i.e., the function W (t)f satisfies a semigroup law:
Introduction Functions of bounded variation, usually denoted by BV, have had and have an important role in several problems of calculus of variations. The main features that make BV functions suitable
More informationSplitting methods with boundary corrections
Splitting methods with boundary corrections Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017 Strang s paper, SIAM J. Numer. Anal., 1968 S (5)
More informationThe Calderon-Vaillancourt Theorem
The Calderon-Vaillancourt Theorem What follows is a completely self contained proof of the Calderon-Vaillancourt Theorem on the L 2 boundedness of pseudo-differential operators. 1 The result Definition
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationFractional convexity maximum principle
Fractional convexity maximum principle Antonio Greco Department of Mathematics and Informatics via Ospedale 7, 0914 Cagliari, Italy E-mail: greco@unica.it Abstract We construct an anisotropic, degenerate,
More informationGreen s Functions and Distributions
CHAPTER 9 Green s Functions and Distributions 9.1. Boundary Value Problems We would like to study, and solve if possible, boundary value problems such as the following: (1.1) u = f in U u = g on U, where
More informationTD M1 EDP 2018 no 2 Elliptic equations: regularity, maximum principle
TD M EDP 08 no Elliptic equations: regularity, maximum principle Estimates in the sup-norm I Let be an open bounded subset of R d of class C. Let A = (a ij ) be a symmetric matrix of functions of class
More informationPart 1 Introduction Degenerate Diffusion and Free-boundaries
Part 1 Introduction Degenerate Diffusion and Free-boundaries Columbia University De Giorgi Center - Pisa June 2012 Introduction We will discuss, in these lectures, certain geometric and analytical aspects
More informationAsymptotic behavior of infinity harmonic functions near an isolated singularity
Asymptotic behavior of infinity harmonic functions near an isolated singularity Ovidiu Savin, Changyou Wang, Yifeng Yu Abstract In this paper, we prove if n 2 x 0 is an isolated singularity of a nonegative
More informationCHENCHEN MOU AND YINGFEI YI
INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN CHENCHEN MOU AND YINGFEI YI Abstract. In this paper, we study interior regularity properties for the regional fractional Laplacian operator. We obtain
More informationRichard F. Bass Krzysztof Burdzy University of Washington
ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann
More informationNon-stationary Friedrichs systems
Department of Mathematics, University of Osijek BCAM, Bilbao, November 2013 Joint work with Marko Erceg 1 Stationary Friedrichs systems Classical theory Abstract theory 2 3 Motivation Stationary Friedrichs
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationON THE SPECTRUM OF TWO DIFFERENT FRACTIONAL OPERATORS
ON THE SPECTRUM OF TWO DIFFERENT FRACTIONAL OPERATORS RAFFAELLA SERVADEI AND ENRICO VALDINOCI Abstract. In this paper we deal with two nonlocal operators, that are both well known and widely studied in
More informationWave propagation in discrete heterogeneous media
www.bcamath.org Wave propagation in discrete heterogeneous media Aurora MARICA marica@bcamath.org BCAM - Basque Center for Applied Mathematics Derio, Basque Country, Spain Summer school & workshop: PDEs,
More informationT. Godoy - E. Lami Dozo - S. Paczka ON THE ANTIMAXIMUM PRINCIPLE FOR PARABOLIC PERIODIC PROBLEMS WITH WEIGHT
Rend. Sem. Mat. Univ. Pol. Torino Vol. 1, 60 2002 T. Godoy - E. Lami Dozo - S. Paczka ON THE ANTIMAXIMUM PRINCIPLE FOR PARABOLIC PERIODIC PROBLEMS WITH WEIGHT Abstract. We prove that an antimaximum principle
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationBoundary problems for fractional Laplacians
Boundary problems for fractional Laplacians Gerd Grubb Copenhagen University Spectral Theory Workshop University of Kent April 14 17, 2014 Introduction The fractional Laplacian ( ) a, 0 < a < 1, has attracted
More informationSome Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces
Journal of Convex nalysis Volume 1 (1994), No. 2, 135 142 Some Remarks bout the Density of Smooth Functions in Weighted Sobolev Spaces Valeria Chiadò Piat Dipartimento di Matematica, Politecnico di Torino,
More informationR. M. Brown. 29 March 2008 / Regional AMS meeting in Baton Rouge. Department of Mathematics University of Kentucky. The mixed problem.
mixed R. M. Department of Mathematics University of Kentucky 29 March 2008 / Regional AMS meeting in Baton Rouge Outline mixed 1 mixed 2 3 4 mixed We consider the mixed boundary value Lu = 0 u = f D u
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationSome recent results on controllability of coupled parabolic systems: Towards a Kalman condition
Some recent results on controllability of coupled parabolic systems: Towards a Kalman condition F. Ammar Khodja Clermont-Ferrand, June 2011 GOAL: 1 Show the important differences between scalar and non
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationOn the Solvability Conditions for a Linearized Cahn-Hilliard Equation
Rend. Istit. Mat. Univ. Trieste Volume 43 (2011), 1 9 On the Solvability Conditions for a Linearized Cahn-Hilliard Equation Vitaly Volpert and Vitali Vougalter Abstract. We derive solvability conditions
More informationThe Dirichlet-to-Neumann operator
Lecture 8 The Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator plays an important role in the theory of inverse problems. In fact, from measurements of electrical currents at the surface
More informationNon-homogeneous semilinear elliptic equations involving critical Sobolev exponent
Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent Yūki Naito a and Tokushi Sato b a Department of Mathematics, Ehime University, Matsuyama 790-8577, Japan b Mathematical
More informationPotential theory of subordinate killed Brownian motions
Potential theory of subordinate killed Brownian motions Renming Song University of Illinois AMS meeting, Indiana University, April 2, 2017 References This talk is based on the following paper with Panki
More informationREACTION-DIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II - CYLINDRICAL-TYPE DOMAINS. Henri Berestycki and Luca Rossi
Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 REACTION-DIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II - CYLINDRICAL-TYPE
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationEXISTENCE OF SOLUTIONS TO P-LAPLACE EQUATIONS WITH LOGARITHMIC NONLINEARITY
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 87, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS
More informationRegularity estimates for fully non linear elliptic equations which are asymptotically convex
Regularity estimates for fully non linear elliptic equations which are asymptotically convex Luis Silvestre and Eduardo V. Teixeira Abstract In this paper we deliver improved C 1,α regularity estimates
More informationViscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationConservation law equations : problem set
Conservation law equations : problem set Luis Silvestre For Isaac Neal and Elia Portnoy in the 2018 summer bootcamp 1 Method of characteristics For the problems in this section, assume that the solutions
More informationTHE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM
THE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM J. GARCÍA-AZORERO, J. J. MANFREDI, I. PERAL AND J. D. ROSSI Abstract. We consider the natural Neumann boundary condition
More information