Suitable Solution Concept for a Nonlinear Elliptic PDE Noureddine - PowerPoint PPT Presentation

Suitable Solution Concept for a Nonlinear Elliptic PDE Noureddine Igbida Institut de recherche XLIM, UMR-CNRS 6172 Universit´ e de Limoges 87060 Limoges, France Colloque EDP-Normandie 2011 Rouen, 25-26 Octobre 2011 N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Collaboration F. Andreu, J. Mazon and J. Toledo : University of Valencia (Spain) Soma Safimba and Stanislas Ouaro : University of Ouaga-Dougou (Burkina Faso) Fahd Karami : University of Essaouira (Maroc)  β ( u ) − ∇ · a ( x , ∇ u ) ∋ µ in Ω  γ ( u ) + a ( x , ∇ u ) · η ∋ ψ on ∂ Ω  Ω ⊂ R N is a bounded regular domain ∂ Ω =: Γ is the boundary γ and β are maximal montone graphes in R × R a is a Leray-Lions type operator ; i.e. a : Ω × R N → R N is Carath´ eodory and satisfies a ( x , ξ ) · ξ � λ | ξ | p , λ > 0 , 1 < p < ∞ . H1 : ′ | a ( x , ξ ) | ≤ σ ( g ( x ) + | ξ | p − 1 ) , g ∈ L p H2 : (Ω) , σ > 0 . H3 : ( a ( x , ξ ) − a ( x , η )) · ( ξ − η ) > 0 . Questions Existence and uniqueness of the solution in the case where µ ∈ L p ′ (Ω) , ψ ∈ L p ′ (Γ) . µ ∈ L 1 (Ω) , ψ ∈ L 1 (Γ) . µ and ψ are Radon measures (diffuse). N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Aim of the talk The case D ( β ) � = R . µ ∈ M b (Ω) is diffuse  β ( u ) − ∇ · a ( x , ∇ u ) ∋ µ in Ω  u = 0 on ∂ Ω  ψ, µ ∈ L p ′  β ( u ) − ∇ · a ( x , ∇ u ) ∋ µ in Ω  γ ( u ) + a ( x , ∇ u ) · η ∋ ψ on ∂ Ω  N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Plan of the talk I - Examples, definitions, reminders II - Examples of nonexistence and our main results III - Main ideas of the proofs N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Examples, definitions, reminders N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Examples Nonlinear and linear diffusion : a ( x , ∇ u ) = ∇ u : Laplace operator a ( x , ∇ u ) = |∇ u | p − 1 ∇ u : p − Laplace operator, 1 < p < ∞ 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 : Nonhomogeneous Neumann boundary condition D ( γ ) = { 0 } : Dirichlet boundary condition Applications : Heat equation, nonlinear diffusion in porous medium Stefan problem, Hele-Shaw problem, Obstacle problem ..... N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Solutions (Dirichlet boundary condition) Weak solution : is a couple ( u , z ) ∈ W 1 , p (Ω) × L 1 (Ω) such that 0 z ∈ β ( u ) , L N − a.e. in Ω , and in D ′ (Ω) . −∇ · a ( x , ∇ u ) = µ − z , 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 (Ω) , T k u ∈ W 1 , p (Ω) , for any k � 0 , z ∈ β ( u ) a.e. in Ω and 0 −∇ · h ( u ) a ( x , Du ) + h ( u ) z = h ( u ) µ − h ′ ( u ) a ( x , ∇ u ) · ∇ u , in D ′ (Ω) , for any h ∈ C c ( R ) , and � |∇ u | p dx = 0 . lim n →∞ [ n ≤| u |≤ n +1] Entropic solution : is a couple of measurable function ( u , z ) such that z ∈ L 1 (Ω) , T k u ∈ W 1 , p (Ω) , for any k � 0 , z ∈ β ( u ) a.e. in Ω and 0 � � � a ( x , ∇ u ) · ∇ T k ( u − ξ ) + z T k ( u − ξ ) ≤ µ T k ( u − ξ ) , Ω Ω Ω for any k � 0 and ξ ∈ W 1 , p (Ω) ∩ L ∞ (Ω) . 0 N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Remarks In the case of Neumann boundary condition one needs to take the test function ξ ∈ W 1 , p (Ω) . And, for renormalized/entropic solution, one needs to work in T 1 , p tr (Ω) . In many cases, an enropic/renormalized solution is a solution in the sense of (Ω) × L 1 (Ω) such that z ∈ β ( u ) , L N − a.e. in Ω , and distribution : ( u , z ) ∈ W 1 , 1 0 in D ′ (Ω) . −∇ · a ( x , ∇ u ) = µ − z , 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. In the case where a ( x , ξ ) = ξ, a solution in the sense of distribution is unique. A weak solution is an enropic/renormalized solution. If ( u , w ) is an enropic/renormalized solution and u ∈ L ∞ (Ω) , then ( u , z ) is a weak solution. N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Existence and uniqueness result Dirichlet boundary condition  β ( u ) − ∇ · a ( x , ∇ u ) ∋ µ in Ω  u = 0 on ∂ Ω  µ ∈ W − 1 , p ′ (Ω) : (P) is well posed in the sense of weak solution. µ ∈ L 1 (Ω) : (P) is well posed in the sense of renormalized solution. µ ∈ M b (Ω) : (P) is well posed in the sense of renormalized solution if D ( β ) = R µ is diffusive. Recall that, a Radon measure µ is said to be diffuse with respect to the capacity W 1 , p (Ω) ( p − capacity for short) if 0 µ ( E ) = 0 for every set E such that cap p ( E , Ω) = 0 . The p − capacity of every subset E with respect to Ω is defined as : � � � |∇ u | p dx ; u ∈ W 1 , p cap p ( E , Ω) = inf (Ω) , u � 0 , s.t. u = 1 a.e. E . 0 Ω The set of diffuse measures is denoted by M p b (Ω) . N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Existence and uniqueness result General boundary condition  β ( u ) − ∇ · a ( x , ∇ u ) ∋ µ in Ω  γ ( u ) + a ( x , ∇ u ) · η ∋ ψ on ∂ Ω  µ, ψ ∈ L p ′ : (P) is well posed in the sense of weak solution when (A) D ( β ) = R (B) ψ ≡ 0 µ, ψ ∈ L 1 : (P) is well posed in the sense of of renormalized solution when (A) D ( β ) = R (B) ψ ≡ 0 µ ∈ M b (Ω) : (P) is well posed in the sense of renormalized solution if D ( β ) = R µ is diffusive. Remark The non existence of standard solution appears in the following cases : D ( β ) � = R Dirichlet boundary condition and Radon measure µ (diffuse) D ( β ) � = R Non homogeneous Neumann boundary condition. N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Bibliography Very large litterature : Dirichlet and homogeneous Neumann boundary condition : B´ enilan, Brezis, Boccardo, T Gallouet, Murat, Blanchard, Crandall, Redouan, Guib´ e, Porreta, Dal Maso, Orsina, Prignet, NI, Mazon, Toledo, Andreu ..... Nonhomogeneous 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) : NI, 2002/06. General cases : F. Andreu, J. Mazon, J. Toledo, NI : 2008-2011 N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Pioneering works In the particular case a ( x , ξ ) = ξ, the problem reads  − ∆ u + β ( u ) ∋ µ in Ω  ∂ η u + γ ( u ) ∋ ψ on ∂ Ω .  Id R , γ a maximal monotone graph and µ, ψ ∈ L 2 H. Brezis: β = I Brezis-Strauss : µ ∈ L 1 (Ω) , ψ ≡ 0 and γ , β continuous nondecreasing functions from R into R with β ′ � ǫ > 0. Ph. B´ enilan, M. G. Crandall and P. Sacks : γ and β maximal monotone graphs in R 2 such that 0 ∈ γ (0) ∩ β (0) , µ ∈ L 1 and ψ ≡ 0 . N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Pioneering works For the general case � β ( u ) − ∇ · a ( x , ∇ u ) ∋ µ dans Ω γ ( u ) − a ( x , ∇ u ) · η ∋ ψ sur ∂ Ω 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. Murat, 1994 : Dirichlet boundary condition, µ a Radon measure. B´ enilan, 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. Blanchard and Murat, 1997 : ”parabolic case”. F. Andreu, J. Mazon, J. Toledo and NI : existence and uniqueness of weak (or entropy/renormalized) solutions in the case µ and ψ ∈ L 1 ( ∂ Ω) , in the cases (A) D ( β ) = R (B) ψ ≡ 0 µ and ψ are two Radon diffuse measures , in the case (A) D ( β ) = R N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Examples of nonexistence and our main results N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE

Aim Our aim here is to treat the case where D ( β ) � = R . Examples Signorini problem (elasticity), unilateral constraint :  ∅ if r < 0  β ( r ) = ] − ∞ , 0] if r = 0 0 if r > 0 ,  Optimal control problem, modeling of semipermeability : ∅ if r < m   ] − ∞ , 0] if r = m    β ( r ) = 0 if r ∈ ] m , M [ [0 , + ∞ [ if r = M     ∅ if r > M , where m < 0 < M . N. Igbida Suitable Solution Concept for a Nonlinear Elliptic PDE