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 Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 1 / 43
Structure of the thesis PART I: Integro-differential equations ( ) Lu(x) = PV u(x) u(x + y) K(y)dy R n PART II: Regularity of stable solutions to elliptic equations u = λ f (u) in Ω R n PART III: Isoperimetric inequalities with densities Ω Ω n 1 n B 1 B 1 n 1 n Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 2 / 43
PART I 1. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, [J. Math. Pures Appl. 14] 2. The Pohozaev identity for the fractional Laplacian, [ARMA 14] 3. Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, [Comm. PDE 14] 4. Boundary regularity for fully nonlinear integro-differential equations, Preprint. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 3 / 43
PART II 5. Regularity of stable solutions up to dimension 7 in domains of double revolution, [Comm. PDE 13] 6. The extremal solution for the fractional Laplacian, [Calc. Var. PDE 14] 7. Regularity for the fractional Gelfand problem up to dimension 7, [J. Math. Anal. Appl. 14] Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 4 / 43
PART III 8. Sobolev and isoperimetric inequalities with monomial weights, [J. Differential Equations 13] 9. Sharp isoperimetric inequalities via the ABP method, Preprint. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 5 / 43
PART I: Integro-differential equations Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 6 / 43
Nonlocal equations Linear elliptic integro-differential operators: ( ) Lu(x) = PV u(x) u(x + y) K(y)dy, R n with K 0, K(y) = K( y), and R n min ( 1, y 2) K(y)dy <. Brownian motion 2nd order PDEs Lévy processes Integro-Differential Equations Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 7 / 43
Expected payoff Brownian motion u = 0 in Ω u = φ on Ω u(x) = E ( φ(x τ ) ) (expected payoff) X t = Random process, X 0 = x τ = first time X t exits Ω Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 8 / 43
Expected payoff Brownian motion u = 0 u = φ in Ω on Ω Lévy processes Lu = 0 u = φ in Ω in R n \ Ω u(x) = E ( φ(x τ ) ) (expected payoff) X t = Random process, X 0 = x τ = first time X t exits Ω Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 8 / 43
More equations from Probability Distribution of the process X t Fractional heat equation t u + Lu = 0 Expected hitting time / running cost Controlled diffusion Optimal stopping time Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability Distribution of the process X t Fractional heat equation t u + Lu = 0 Expected hitting time / running cost Dirichlet problem Lu = f (x) u = 0 in Ω in R n \ Ω Controlled diffusion Optimal stopping time Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability Distribution of the process X t Fractional heat equation t u + Lu = 0 Expected hitting time / running cost Controlled diffusion Dirichlet problem Lu = f (x) u = 0 in Ω in R n \ Ω Fully nonlinear equations sup L α u = 0 α A Optimal stopping time Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability Distribution of the process X t Fractional heat equation t u + Lu = 0 Expected hitting time / running cost Controlled diffusion Dirichlet problem Lu = f (x) u = 0 in Ω in R n \ Ω Fully nonlinear equations sup L α u = 0 α A Optimal stopping time Obstacle problem Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
The fractional Laplacian Most canonical example of elliptic integro-differential operator: ( ) s u(x) u(x + y) u(x) = c n,s PV R y n+2s dy, s (0, 1). n Notation justified by ( ) s u(ξ) = ξ 2s û(ξ), ( ) s ( ) t = ( ) s+t. It corresponds to stable and radially symmetric Lévy process. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 10 / 43
Stable Lévy processes Special class of Lévy processes: stable processes ( )a(y/ y ) Lu(x) = PV u(x) u(x + y) R y n+2s dy n Very important and well studied in Probability These are processes with self-similarity properties (X t t 1/α X 1 ) Central Limit Theorems stable Lévy processes a(θ) is called the spectral measure (defined on S n 1 ). Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 11 / 43
Why studying nonlocal equations? Nonlocal equations are used to model (among others): Prices in Finance (since the 1990 s) Anomalous diffusions (Physics, Ecology, Biology): u t + Lu = f (x, u) Also, they arise naturally when long-range interactions occur: Image Processing Relativistic Quantum Mechanics + m Boltzmann equation Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 12 / 43
Why studying nonlocal equations? Still, these operators appear in: Fluid Mechanics (surface quasi-geostrophic equation) Conformal Geometry Finally, all PDEs are limits of nonlocal equations (as s 1). Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 13 / 43
Important works Works in Probability 1950-2014 (Kac, Getoor, Bogdan, Bass, Chen,...) Fully nonlinear equations: Caffarelli-Silvestre 07-10 [CPAM, Annals, ARMA] Reaction-diffusion equations u t + Lu = f (x, u) Obstacle problem, free boundaries Nonlocal minimal surfaces, fractional perimeters Math. Physics: (Lieb, Frank,...) [JAMS 08], [Acta Math. 13] Fluid Mech.: Caffarelli-Vasseur [Annals 10], [JAMS 11] Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 14 / 43
The classical Pohozaev identity u = f (u) in Ω u = 0 on Ω, Theorem (Pohozaev, 1965) { n F (u) n 2 } u f (u) = 1 2 2 Ω Follows from: For any function u with u = 0 on Ω, Ω (x u) u = 2 n 2 Ω u u + 1 2 And this follows from the divergence theorem. Ω Ω ( ) 2 u (x ν)dσ ν ( ) 2 u (x ν)dσ ν Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 15 / 43
The classical Pohozaev identity Applications of the identity: Nonexistence of solutions: critical exponent u = u n+2 n 2 Unique continuation from the boundary Monotonicity formulas Concentration-compactness phenomena Radial symmetry Stable solutions: uniqueness results, H 1 interior regularity Other: Geometry, control theory, wave equation, harmonic maps, etc. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 16 / 43
Pohozaev identities for ( ) s Assume ( ) s u C in Ω u = 0 in R n \ Ω, (+ some interior regularity on u) Theorem (R-Serra 12; ARMA) If Ω is C 1,1, (x u) ( ) s u = 2s n 2 Ω Ω u ( ) s u Γ(1 + s)2 2 Ω ( u d s (x) ) 2 (x ν) Here, Γ is the gamma function. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 17 / 43
Remark 1 2 Ω ( ) 2 u (x ν) ν Γ(1 + s) 2 2 Ω ( u d s ) 2 (x ν) u d s Ω plays the role that u ν plays in 2nd order PDEs Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 18 / 43
Pohozaev identities for ( ) s Changing the origin in our identity, we find Thus, Corollary Ω u xi ( ) s u = Under the same hypotheses as before ( ) s u v xi Ω Γ(1 + s)2 2 Ω ( u d s ) 2 νi = u xi ( ) s v + Γ(1 + s) 2 Ω Ω u d s v d s ν i Note the contrast with the nonlocal flux in the formula Ω ( )s w = R n \Ω Ω... Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 19 / 43
Ideas of the proof 1 u λ (x) = u(λx), λ > 1, 2 Ω star-shaped Ω Ω (x u)( ) s u = d dλ (x u)( ) s u = 2s n 2 w = ( ) s 2 u λ=1 + Ω u λ ( ) s u u( ) s u + 1 d Ω 2 dλ λ=1 + R n w λ w 1/λ, 3 Analyze very precisely the singularity of ( ) s 2 u along Ω, and compute. 4 Deduce the result for general C 1,1 domains. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 20 / 43
Fractional Laplacian: Two explicit solutions 1. u(x) = (x + ) s satisfies ( ) s u = 0 in (0, + ). 2. Explicit solution by [Getoor, 1961]: ( ) s u = 1 in B 1 u = 0 in R n \ B 1 = u(x) = c( 1 x 2) s They are C inside Ω, but C s (Ω) and not better! In both cases, they are comparable to d s, where d(x) = dist(x, R n \ Ω). Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 21 / 43
Boundary regularity: First results ( ) s u = g in Ω u = 0 in R n \Ω, Then, u C s (Ω) C g L. Moreover, Theorem (R-Serra 12; J. Math. Pures Appl.) Ω bounded and C 1,1 domain. Then, u/d s C γ (Ω) C g L for some small γ > 0, where d is the distance to Ω. Proof: Can not do odd reflection! (boundary behavior different from interior!) Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 22 / 43
Boundary for integro-differential operators? ( ) s u = g(x) in Ω u = 0 in R n \Ω, = u/d s C γ (Ω). We answer an open question: What about boundary regularity for more general operators of order 2s? ( ) Lu(x) = PV u(x) u(x + y) K(y)dy R n Is it true that u/d s is Hölder continuous? At least bounded? We answer this for linear and also for fully nonlinear equations Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 23 / 43
Fully nonlinear integro-differential equations Let us consider solutions to Iu = f in Ω where I is a fully nonlinear operator like u = 0 in R n \Ω, Iu(x) = sup L α u(x) α (controlled diffusion) Here, all L α L for some class of linear operators L. The class L is called the ellipticity class. (When L α are 2nd order operators, we have F (D 2 u) = f (x) in Ω) Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 24 / 43
Interior regularity: Was developed by Caffarelli and Silvestre in 2007-2010 (CPAM, Annals, ARMA) They established: Krylov-Safonov, Evans-Krylov, perturbative theory, etc. The reference ellipticity class of Caffarelli-Silvestre is L 0, with kernels Boundary regularity: λ K(y) Λ y n+2s y n+2s We establish boundary regularity for the class L, with kernels K(y) = a (y/ y ) y n+2s, λ a(θ) Λ L = Subclass of L 0, corresponding to stable Lévy processes Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 25 / 43
Boundary regularity for fully nonlinear equations Let I (u, x) be a fully nonlinear operator elliptic w.r.t. L, and I (u, x) = f (x) in Ω u = 0 in R n \Ω, Theorem (R-Serra; preprint 14) If Ω is C 1,1, then any viscosity solution satisfies u/d s C s ɛ (Ω) C f L (Ω) for all ɛ > 0 Even for ( ) s we improve our previous results! Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 26 / 43
Novelty: We obtain higher regularity for u/d s! The exponent s ɛ is optimal for f L Also, it cannot be improved if a L (S n 1 ) Very important: L is the good class for boundary regularity! The class L 0 is too large for fine boundary regularity There exist positive numbers 0 < β 1 < s < β 2 such that I 1 (x + ) β1 0, I 2 (x + ) β2 0 in {x > 0} Solutions are not even comparable near the boundary! Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 27 / 43
Main steps of the proof of u/d s C s ɛ (Ω): 1 Bounded measurable coefficients = u/d s C γ (Ω) 2 Blow up the equation at x Ω + compactness argument. 3 Liouville theorem in half-space Advantages of the method: It allows us to obtain higher regularity of u/d s, also in the normal direction! After blow up, you do not see the geometry of the domain Also non translation invariant equations Discontinuous kernels a L (S n 1 ) Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 28 / 43
PART II: Regularity of stable solutions to elliptic equations Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 29 / 43
Regularity of minimizers Classical problem in the Calculus of Variations: Regularity of minimizers Example in Geometry: Regularity of hypersurfaces in R n which minimize the area functional. These hypersurfaces are smooth if n 7 In R 8 the Simons cone minimizes area and has a singularity at x = 0 As we will see, the same happens for other nonlinear PDE in bounded domains. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 30 / 43
Regularity of minimizers Open problem: u = f (u) in Ω R n u = 0 on Ω, u local minimizer (or stable solution) & n 9 = u L? In R 10, u(x) = log 1 x 2 is a stable solution in B 1 f (u) = λe u or f (u) = λ(1 + u) p & n 9 [Crandall-Rabinowitz 75] Ω = B 1 & n 9 [Cabré-Capella 06] n 4 [n 3 Nedev 00; n 4 Cabré 10] Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 31 / 43
The extremal solution If f (u) λf (u), then there is λ (0, + ) s.t. For 0 < λ < λ, there is a bounded solution u λ. For λ > λ, there is no solution. For λ = λ, u (x) = lim λ λ u λ(x) is a weak solution, called the extremal solution. Moreover, it is stable. Question: Is the extremal solution bounded? [Brezis-Vázquez 97] Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 32 / 43
Our work We have studied the regularity of stable solutions to u = f (u) in Ω u = 0 on Ω, [Comm. PDE 13] Thm: L for n 7 & Ω of double revolution ( ) s u = f (u) in Ω u = 0 in R n \ Ω, [Calc. Var. PDE 13] [J. Math. Anal. Appl. 13] Thm: L and H s bounds in general domains; Thm: optimal regularity for f (u) = λe u in x i -symmetric domains Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 33 / 43
Sobolev inequalities with weights When studying u = f (u), we needed { s α u s 2 + t β u t 2} ds dt C = u L q (Ω)? q(α, β) =? Ω After a change of variables, we want ( ) 1/q ( ) 1/2 u q x1 a x2 b dx 1 dx 2 C u 2 x1 a x2 b dx 1 dx 2, q = q(a, b) Ω Ω Thus, we want Sobolev inequalities with weights ( ) 1/q ( ) 1/p u q w(x)dx C u p w(x)dx, w(x) = x A1 1 xn An R n R n Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 34 / 43
PART III: Isoperimetric inequalities with densities Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 35 / 43
Sobolev inequalities with weights Theorem (Cabré-R; J. Differential Equations 13) Let A i 0, Let 1 p < D. Then, with q = pd/(d p). w(x) = x A1 1 x An n, D = n + A 1 + + A n. ( ) 1/q ( ) 1/p u q w(x)dx C p,a u p w(x)dx, R n R n To prove the result, we establish a new weighted isoperimetric inequality. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 36 / 43
Isoperimetric inequalities with monomial weights Theorem (Cabré-R; J. Differential Equations 13) Let A i > 0, w(x) and D as before, and Σ = {x R n : x 1,..., x n > 0}. Then, for any E Σ, P w (E) w(e) D 1 D P w (B 1 Σ). w(b 1 Σ) D 1 D We denoted the weighted volume and perimeter w(e) = w(x)dx P w (E) = w(x)ds. E Σ E Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 37 / 43
Isoperimetric inequalities with weights This type of isoperimetric inequalities have been widely studied: w(x) = e x 2 [Borell; Invent. Math. 75] Existence and regularity of minimizers (Pratelli, Morgan,...) log-convex radial densities w( x ) [Figalli-Maggi 13], [Chambers 14] with w(x) = e x 2, w(x) = x α, or other particular weights... Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 38 / 43
Isoperimetric inequalities in cones In general cones Σ, a well known result is the following: Theorem (Lions-Pacella 90) Let Σ be any open convex cone in R n. Then, for any E Σ, Σ E E n n 1 Σ B 1. B 1 Σ n n 1 Important: Only the perimeter inside Σ is counted Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 39 / 43
New isoperimetric inequalities weights Theorem (Cabré-R-Serra; preprint 13) Let Σ be any convex cone in R n. Assume w(x) homogeneous of degree α 0, & w 1/α concave in Σ. Then, for any E Σ, P w (E) w(e) D 1 D P w (B 1 Σ). w(b 1 Σ) D 1 D Recall w(e) = w(x)dx P w (E) = w(x)ds. E Σ E Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 40 / 43
Comments Minimizers are radial, while w(x) is not! When w 1 we recover the result of Lions-Pacella (with new proof!). We can also treat anisotropic perimeters P w,h (E) = H(ν) w(x)ds. Σ E w 1 = new proof of the Wulff theorem. Some examples of weights are w(x) = dist(x, Σ) α, w(x) = x + y, w(x) = xyz x + y + z,... Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 41 / 43
The proof The proof uses the ABP technique applied to an appropriate PDE When w 1, the idea goes back to the work [Cabré 00] (for the classical isoperimetric inequality) Here, we need to consider a linear Neumann problem in E Σ involving the operator w 1 div(w u) Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 42 / 43
The end Thank you! Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 43 / 43