Suitable Solution Concept for a Nonlinear Elliptic PDE

Transcription

1 Suitable Solution Concept for a Nonlinear Elliptic PDE Noureddine Igbida Institut de recherche XLIM, UMR-CNRS 6172 Université de Limoges Limoges, France Colloque EDP-Normandie 2011 Rouen, Octobre 2011

2 Program (S β,γ µ,ψ ) { β(u) a(x, u) µ γ(u) a(x, u) η ψ dans sur where R N is a bounded regular domain =: Γ is the boundary γ and β are maximal montone graphes in R R a a Leray-Lions type operator ; i.e. a : R N R N is Carathéodory and satisfies H1 a(x, ξ) ξ λ ξ p, λ > 0. Questions H2 a(x, ξ) σ(g(x) + ξ p 1 ), g L p (), σ > 0. H3 (a(x, ξ) a(x, η)) (ξ η) > 0. Existence and uniqueness of the solution in the case where µ L p (), ψ L p (Γ). µ L 1 (), ψ L 1 (Γ). µ and ψ are Radon measures. Collaborations : F. Andreu, J. Mazon and J. Toledo : University of Valencia (Spain) Soma Safimba and Stanislas Ouaro : University of Ouga Dougou (Burkina Faso) Fahd Karami : University of Essaouira (Maroc)

3 Examples Nonlinear and linear diffusion : a(x, u) = u : Laplace operator a(x, u) = u p 1 u : p Laplace operator. β(r) = r α r : porous medium equation β(r) = (r 1) + (r 1) : Stefan problem β = H (Heaviside graph) : Hele-Shaw problem β = H 1 : Obstacle problem Boundary condition : γ 0 : Non homogeneous Neumann boundary condition D(γ) = {0} : Dirichlet boundary condition Applications : Heat equation, nonlinear diffusion in porous medium Stefan problem, Hele-Shaw problem, Obstacle problem...

4 Bibliography Very large litterature : Dirichlet and Neumann homogène : Bénilan, Brezis, Boccardo, T Gallouet, Murat, Blanchard, Crandall, Redouan, Guibé, Porreta, Dal Maso, Orsina, Prignet... Non homogeneous Neumann boundary condition Laplacien with β 0 and γ continuous : J. Hulshof, 1987 Laplacien with γ and β continuous in R : N. Kenmochi, 1990 Laplacien with γ 0 and β continuous (not everywhere defined) : N. Igbida, 2002, 2006.

5 Pioneering works In the particular case a(x, ξ) = ξ, the problem (S β,γ µ,ψ ) reads (L β,γ µ,ψ ) u + β(u) µ ηu + γ(u) ψ in on. H. Brezis: β = Id R, γ a maximal monotone graph and ψ L 2 () Brezis-Strauss : µ L 1 (), ψ 0 and γ, β continuous nondecreasing functions from R into R with β ɛ > 0. γ and β maximal monotone graphs in R 2 such that 0 γ(0) β(0): Ph. Bénilan, M. G. Crandall and P. Sacks.

6 Pioneering works For the general case (S β,γ µ,ψ ) { β(u) a(x, u) µ γ(u) a(x, u) η ψ dans sur 90ies truncation and renormalization allowed to characterize the solutions intrinsically, in a way acceptable for the PDE community and became classical in a few years Boccardo and Gallouet, 1992 : Dirichlet boundary condition, β 0 and µ a Radon measure. Lions and Murat : Bénilan, Boccardo, Gallouet, Gariepy, Pierre and Vazquez, 1995 : Dirichlet boundary condition and µ L 1 Dal Maso, Murat, Orsina and Prignet, 1999 : Dirichlet boundary condition and µ a Radon measure. F. Andreu, J. Mazon, J. Toledo and NI : we extend the results of Ph. Bénilan, M. G. Crandall and P. Sacks by proving the existence and uniqueness of weak (or entropy/renormalized) solutions with µ and ψ L 1 ( ), in the cases (A) D(β) = R and, D(γ) = R or div a(x, Du) = p(u), (B) ψ 0 and, D(γ) = R or div a(x, Du) = p(u), µ and ψ are two Radon diffuse measures, in the case (A) D(β) = R and, D(γ) = R or div a(x, Du) = p(u),

7 Solutions (Dirichlet boundary condition) Weak solution : The most standard is the weak solution concept. That is a couple (u, z) W 1,p 0 () L 1 () such that z β(u), L N a.e. in, and a(x, u) = µ z, in D (). In general, if the data are enough regular, weak solutions does not exist. Renormalized solution : is a couple of measurable function (u, z) such that z L 1 (), T k u W 1,p 0 (), for any k 0, z β(u) a.e. in and h(u)a(x, Du) + h(u)z = h(u)µ h (u) a(x, u) u, for any h C c(r), and lim u p dx = 0. n [n u n+1] in D (), Entropic solution : this is an equivalent concept to renormalized solution. That is a couple of measurable function (u, z) such that z L 1 (), T k u W 1,p 0 (), for any k 0, w β(u) a.e. in and a(x, u) T k (u ξ) + z T k (u ξ) µ T k (u ξ), for any k 0 and ξ W 1,p 0 () L ().

8 Remarks In the case of Neumann boundary condition on eneeds to take the test function ξ W 1,p (). And, for renormalized/entropic solution, one needs to work in T 1,p tr (). An enropic/renormalized solution is a solution in the sense of distribution : (u, z) W 1,1 0 () L 1 () such that z β(u), L N a.e. in, and a(x, u) = µ z, in D (). In the case where a(x, ξ) = ξ, there is an equivalence between the entropic solution and the solution in the sense of distribution. In general, a solution in the sense of distribution is not unique. A weak solution is an enropic/renormalized solution. If (u, w) is an enropic/renormalized solution and u L (), then (u, z) is a weak solution.

9 Existence and uniqueness result Dirichlet boundary condition (P) is well posed in the sense of weak solution when µ W 1,p () (P) is well posed in the sense of renormalized solution when µ L 1 (). If D(β) = R, (P) is well posed in the sense of renormalized solution when µ is Radon diffusive Radon measure with respect to the capacity W 1,p 0 (). Recall that, a Radon measure µ is said to be diffuse with respect to the capacity W 1,p 0 () (p capacity for short) if µ(e) = 0 for every set E such that cap p (E, ) = 0. The p capacity of every subset E with respect to is defined as : { } cap p (E, ) = inf u p dx ; u W 1,p 0 (), u 0, s.t. u = 1 a.e. E. The set of diffuse measures is denoted by M p b ().

10 Existence and uniqueness result General boundary condition (P) is well posed in the sense of weak solution when µ W 1,p () and ψ in the case (A) D(β) = R and, D(γ) = R or div a(x, Du) = p(u), (B) ψ 0 and, D(γ) = R or div a(x, Du) = p(u), (P) is well posed in the sense of of renormalized solution when µ L 1 () and ψ L 1 (Γ) in the case (A) D(β) = R and, D(γ) = R or div a(x, Du) = p(u), (B) ψ 0 and, D(γ) = R or div a(x, Du) = p(u), (P) is well posed in the sense of of renormalized solution when µ and ψ are diffuse Radon measure in the case (A) D(β) = R and, D(γ) = R or div a(x, Du) = p(u),

11 Aim Our aim here is to treat the case where Applications : D(β) R. Signorini problem (elasticity), unilateral constraint : if r < 0 β(r) = ], 0] if r = 0 0 if r > 0, Optimal control problem, modeling of semipermeability : where a < 0 < b. if r < a ], 0] if r = a β(r) = 0 if r ]a, b[ [0, + [ if r = b if r > b,

12 Example of non existence Dirichlet boundary condition : Example 1 β be the maximal monotone graph given by { 0 if r < 0 β(r) = [0, ) if r = 0. If µ is nonnegative and (u, z) is a solution of (P), then 0 z T k u + a(x, u) T k u = T k uµ 0, for any k 0. This implies that u 0 and z = µ if µ is a nonnegative Radon measure. Example 2 β : R R continuous nondecreasing, β(0) = 0 and 2 β lim β(t) = + and lim t) β g(t) > 0 t 1 r 1 (1 L. Dupaigne, A. Ponce and A. Porretta prove that there exists a diffuse measure µ with respect to the capacity H 1 () such that the problem { u + β(u) = µ in, u = 0 on. has no weak solution. That measure µ is taken such µ + << H N 2+α, for some 0 < α < 2, where H s denotes the s dimensional Hausdorff measure.

13 Example of non existence Non homogeneous Neumann boundary condition : u + β(u) µ (L γ,0 µ,ψ ) ηu = ψ in on, β is a maximal monotone graph, D(β) = [0, 1] and 0 γ(0) µ L 1 (), µ 0 a.e. in ψ L 1 ( ), ψ 0 a.e. in. If (u, w) is a solution then, 0 u 1 a.e. in 0 Du 2 + zu = µu + ψu 0 u is constant and zv = µv + ψv, v H 1 () L () µ = z a.e. in and ψ 0 a.e. in. nonexistence if 0 > ψ 0 and µ 0.

14 Main results Assume for instance D(β) = [m, M]. Case : Dirichlet boundary condition and diffuse Radon measure data Theorem (NI, S. Safimba and S. Oauro,JDE-2010) For any µ M p b (), the problem { β(u) a(x, u) µ in u = 0 on has a unique solution (u, z) in the sense that (u, z) W 1,p 0 () L 1 (), z β(u) L N a.e. in there exists ν M p b () such that ν LN ν + is concentred on [u = M] ν is concentred on [u = m] for any ξ W 1,p 0 () L () a(x, u). ξdx + zξdx + ξdν = ξdµ. Moreover, we have ν + << µ s [u = M] and ν << µ s [u = m].

15 Main results Assume for instance D(β) = [m, M]. Case : Non homogeneous Neumann boundary condition and L p data Theorem (NI, S. Safimba and S. Oauro,JDE-2010) For any µ L p () and ψ L p ( ), the problem β(u) a(x, u) µ a(x, u) η = ψ in on, has a unique solution (u, z) in the sense that (u, z) W 1,p () L 1 (), z β(u) there exists ν L 1 ( ) such that L N a.e. in ν + is concentred on [u = M] ν is concentred on [u = m] for any ξ W 1,p 0 () L () a(x, u). ξdx + zξdx + ξν dx = ξdµ.

16 Main ideas of the existence proof for weak solution Take an approximation : β β n au sens des graphes β n(u n) a(x, u n) = µ in Aims : S1 : u n converges weakly in W 1,p () S2 : z n converges in L 1 () (at least weakly) S3 : Identification : identification of lim a(x, un) : Minty type arguments n identification of lim βn(un) : monotony, strong/weak convergence of un/βn(un) n Dirichlet boundary condition S1 : Test with u n (+) Poincaré inequality = u n is bounded in W 1,p () S2 : Estimates in L q, for any q 1, = z n converges in L 1 ()-weakly See here that there is no restriction on D(β) Existence of weak solution for any maximal monotone graph β

17 Main ideas of the existence proof for weak solution Take an approximation : β β n au sens des graphes β n(u n) a(x, u n) = µ in Aims : S1 : u n converges weakly in W 1,p () S2 : z n converges in L 1 () (at least weakly) S3 : Identification : identification of lim a(x, un) : Minty type arguments n identification of lim βn(un) : monotony, strong/weak convergence of un/βn(un) n Neumann boundary condition S1 : u n 1 u n is bounded in W 1,p () Additional assumptions (necessary condition) = 1 u n is bounded S2 : Monotone approximation for D(β) = R : β m n := β + 1 m I + 1 n I = z m n converges in L1 (). See here that D(βn m ) = D(β) Existence of weak solution for maximal monotone graph such that D(β) = R

18 Main ideas of the existence proof for renormalized solution Assume we have existence of a weak solution for regular data µ. Take an approximation : µ µ n in L 1 () Main Steps : β(u n) a(x, u n) = µ in S1 : The convergence of u n : the estimates in W 1,p () falls to be true!! T k (u ɛ) T k (u) weakly in W 1,p () a(x, T k (u ɛ)) a(x, T k (u)) weakly in L p () T k (u ɛ) T k (u) a.e in a(x, T k (u ɛ)). T k (u ɛ) a(x, T k (u)) T k (u) a.e in and strongly in L 1 () T k (u ɛ) T k (u) strongly in L p () S2 : The convergence of z n The contraction property = z n converges in L 1 () Existence of a renormalized solution : - Dirichlet boundary condition without any restriction on D(β) - Neuman boundary condition for D(β) = R

19 Diffuse Radon measure data µ is diffuse = µ = f + F, f L 1 (), F L p () N Take an approximation Main Steps : β ε(u ε) a(x, u ε) = f ε + F ε in S1 : The convergence of u n : the estimates in W 1,p () falls to be true!! T k (u ɛ) T k (u) weakly in W 1,p () a(x, T k (u ɛ)) a(x, T k (u)) weakly in L p () T k (u ɛ) T k (u) a.e in a(x, T k (u ɛ)). T k (u ɛ) a(x, T k (u)) T k (u) a.e in and strongly in L 1 () T k (u ɛ) T k (u) strongly in L p () S2 : The convergence of z n z n is bounded in L 1 () z n z in M b () weak Radon-Nykodym decomposition relatively to Lebesgue measure : z = z r + z s The characterization of z r and z s???

20 Characterization of z We have A sequences (v ɛ) ɛ>0 and (z ɛ) ɛ>0 of measurable functions on satisfying z ɛ β ɛ(v ɛ) L N a.e in v ɛ v L N a.e in, z n z = z r + z s in M b () weak β ε β in the sense of graphe The aims To show that u(x) D(β), L N a.e. x? To show that z r (x) β(u(x)), L N a.e. x ν + is concentred on [u = M] and ν is concentred on [u = m]

21 u(x) D(β), L N a.e. x u(x) D(β), L N a.e. x? Lemma Let (β n) n 1 be a sequence of maximal monotone graphs such that β n β in the sense of graphs. We consider (z n) n 1 and (w n) n 1 two sequences of L 1 (), such that w n β n(z n), L N a.e. in, for any n = 1, 2,... If (w n) n 1 is bounded in L 1 () and z n z in L 1 (), then z dom(β) L N a.e. in. The main tool for the proof of this Lemma is the bitting lemma of Chacon Lemma The bitting lemma of Chacon Let R N be an open bounded of R N and (f n) n a bounded sequence in L 1 (). Then there exist f L 1 (), a subsequence (f nk ) k and a sequence of measurable sets (E j ) j, E j, j N with E j+1 E j and lim E j = 0, such that for any j N, f nk f in L 1 ( ) \E j. j +

22 Idea of the proof (w n) n 1 is bounded in L 1 () bitting lemma of Chacon there exist w L 1 (), a subsequence (w nk ) k 1 a sequence of mesurable sets (E j ) j N in such that and E j+1 E j, j N, lim E j = 0 j + j N, w nk w in L 1 (\E j ). z nk z, in L 1 () and so in L 1 (\E j ), j N β in the sense of graphs β nk w β(z), a.e in \E j z dom(β), a.e in \E j. z dom(β), a.e in.

23 characterization of z r and z s We have A sequences (v ɛ) ɛ>0 and (z ɛ) ɛ>0 of measurable functions on satisfying Lemma z ɛ β ɛ(v ɛ), v ɛ v, L N a.e in, z n z = z r + z s, in M b () weak β ε β, in the sense of graphe L N a.e in (+) β ε = j ɛ, j ε is lower semi-continuous functions and v dom(j), L N a.e in, j ɛ(r) 0, ɛ > 0 and j ɛ j as ɛ 0. Assume moreover, that dom(j) = [m 1, m 2 ] [ l 0, l 0 ], z M p b () satisfies lim inf (t v ɛ)ξh 0 (v ɛ)z ɛdx (t v)ξdz, t R, (1) ɛ 0 ξ Cc 1 (), ξ 0, Then z = wl N + z s with z s L N, w j(v) L N a.e in, w L 1 (), z + s is concentrated on [v = m 2 ] and z s is concentrated on [v = m 1 ].

24 Idea of the proof j(t) j ɛ(t) j ɛ(v ɛ) + (t v ɛ)z ɛ, L N a.e in ξh 0 (v ɛ)j(t)dx ξh 0 (v ɛ)j ɛ (v ɛ)dx + (t v ɛ)ξh 0 (v ɛ)z ɛdx. Fatou s Lemma implies that ξh 0 (v)j(t)dx ξh 0 (v)j ɛ (v)dx + lim inf (t v ɛ)ξh 0 (v ɛ)z ɛdx. ɛ 0 The assumption of the lemma implies that ξh 0 (v)j(t)dx ξh 0 (v)j ɛ (v)dx + ξ(t v)dz. Letting ɛ 0, and using again Fatou s Lemma, we get ξh 0 (v)j(t)dx ξh 0 (v)j(v)dx + ξ(t v)dz Then h 0 (v)j(t) h 0 (v)j(v) + (t v)z, in M b (), t R Since [m 1, m 2 ] [ l 0, +l 0 ], we get j(t) j(v) + (t v)z, in M b (), t R Comparing the regular part and the singular part, we obtain j(t) j(v) + (t v)w, L N a.e in, t R and (t v)z s 0 in M b (), t dom(j) This implies that z r (x) β(u(x)), L N a.e. x

25 Bibliography N. Igbida, S. Ouaro and S. Safimba Elliptic Problem Involving Diffuse Measure Data. Accepté dans J. Diff. Equation. F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Obstacle Problems for Degenerate Elliptic Equations with Nonlinear Boundary Conditions. Math. Models Methods Appl. Sci., Vol. 18, No. 11 (2008) 1869?1893. *********************************** F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, L 1 Existence and Uniqueness Results for Quasi-linear Ellipti Equations with Nonlinear Boundary Conditions. Ann. I. H. Poincarém Anal. Non Linéaire 24 (2007), F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Interfaces Free Bound. 8 (2006), F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Degenerate Elliptic Equations with Nonlinear Boundary Conditions and Measures Data. Ann. Scuola Normale Sup. Pisa, Cl. Sci. (5) Vol. VIII (2009), 1-37 F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Renormalized Solutions for Degenerate Elliptic-Parabolic Problems with Nonlinear Dynamical Boundary Conditions. J. Differential Equations, Vo. 244, 11(2008), N. Igbida, From Fast to Very Fast Diffusion in the Nonlinear Heat Equation. Transaction of the AMS, Vo. 361, No. 10 (2009) 5089?5109 N. Igbida, Hele Shaw Problem with Dynamical Boundary Conditions. Jour. Math. Anal. Applications, Vo. 335 No. 2, , 2007.