Homogenization of a quasilinear elliptic problem with nonlinear - PowerPoint PPT Presentation

Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions Patrizia Donato University of Rouen Symposium Homogenization and Multi Scale Analysis Shanghai, October 3-7, 2011 Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 1/59 •

Outline Position of the problem 1 Existence and uniqueness of a solution 2 Homogenization results 3 The quasilinear homogenized matrix 4 Uniqueness of the homogenized problem 5 About the unfolding method 6 Some preliminary results 7 Proofs of the homogenization results 8 Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 2/59 •

Results in collaboration with Bituin Cabarrubias, Ph.D. student under joint tutorage, University of Philippines and University of Rouen, in B. Cabarrubias and P. Donato, Existence and Uniqueness for a Quasilinear Elliptic Problem With Nonlinear Robin Conditions , to appear in Carpathian J. Math., (2) 2011. B. Cabarrubias and P. Donato, Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions, Applicable Analysis, 2011, to appear. Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 3/59 •

lem The problem Aim: To study the homogenization, as ε → 0, of the following quasilinear elliptic problem:  − div( A ε ( x , u ε ) ∇ u ε ) = f in Ω ∗ ε ,    on Γ ε u ε = 0 0 ,    A ε ( x , u ε ) ∇ u ε · n + ε γ τ ε ( x ) h ( u ε ) = g ε ( x ) on Γ ε 1 , where Ω ∗ ε is a periodically perforated domain of R N , N ≥ 2, ∂ Ω ∗ ε is decomposed in two disjoint parts, Γ ε 0 and Γ ε 1 , n is the unit exterior normal to Ω ∗ ε , γ is a real parameter, with γ ≥ 1. Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 4/59

lem The perforated domain We denote by b = ( b 1 , b 2 , . . . , b N ) a basis of R N , N ≥ 2, Y the corresponding reference cell, i.e. Y = { y ∈ R N | y = � N i =1 y i b i , ( y 1 , . . . , y N ) ∈ (0 , 1) N } , S (the reference hole) a compact subset of Y , Y ∗ = Y \ S the perforated reference cell, { ε } be a positive sequence that converges to zero, G = { ξ ∈ R N | ξ = � N Z N } , i =1 k i b i , ( k 1 , . . . , k N ) ∈ Z Ω a bounded open set of R N . The perforated domain Ω ∗ ε is defined by � Ω ∗ ε = Ω \ S ε , where S ε = ε ( ξ + S ) . ξ ∈ G Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 5/59

lem Figure 1. The perforated domain Ω ∗ ε . We assume that S has a Lipschitz continuous boundary with a finite number of connected components. Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 6/59

lem The two components of the boundary As in the unfolding periodic method, we set � � ε ( ξ + ¯ Λ ε = Ω \ � Ω ε = interior { Y ) } and Ω ε , ξ ∈ Ξ ε where Ξ ε = { ξ ∈ G , ε ( ξ + Y ) ⊂ Ω } . By construction, � Ω ε is the interior of the largest union of ε ( ξ + ¯ Y ) cells such that ε ( ξ + Y ) ⊂ Ω, while Λ ε is the subset of Ω from the ε ( ξ + ¯ Y ) cells intersecting ∂ Ω. We consider the corresponding perforated sets � ε = � ε \ � Ω ∗ Λ ∗ ε = Ω ∗ Ω ∗ Ω ε \ S ε and ε and we decompose the boundary of the perforated domain Ω ∗ ε as 1 = ∂ � ∂ Ω ∗ ε = Γ ε 0 ∪ Γ ε Γ ε Ω ∗ Γ ε 0 = ∂ Ω ∗ ε \ Γ ε 1 , where ε ∩ ∂ S ε and 1 , 1 is the boundary of the set of holes included in � so that Γ ε Ω ε . Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 7/59

lem Figure 2. The perforated domain Ω ∗ ε and its boundary ∂ Ω ∗ ε = Γ ε 0 ∪ Γ ε 1 . In the figure, Ω ε is the dark perforated part, 1 is the boundary of the holes contained in � Γ ε Ω ε , Λ ∗ ε is the remaining perforated part, Γ ε 0 is the union of ∂ Ω and the boundary of the holes in Λ ∗ ε . Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 8/59

lem Assumptions on the data Concerning the equation − div( A ε ( x , u ε ) ∇ u ε ) = f , we assume that f ∈ L 2 (Ω), → R N 2 is a matrix field s.t. A : Y × R �−   ( i ) A ( y , t ) is Y - periodic for every t and a Caratheodory function,     i.e. continuous for a.e. y and measurable for every t ;       ( ii ) A ( · , t ) ∈ M ( α, β, Y ) for every t ∈ R , i . e .      A ( y , t ) λ, λ ) ≥ α | λ | 2 , ∀ λ ∈ R N , a.e. in Y .  | A ( y , t ) λ | ≤ β | λ | ,   ( iii ) there exists a function ω : R �→ R such that      - ω is continuous, nondecreasing and ω ( t ) > 0 ∀ t > 0 ,       - | A ( y , t ) − A ( y , t 1 ) | ≤ ω ( | t − t 1 | ) for a.e. y and t � = t 1 ,    � r   dt   - for any r > 0 , lim ω ( t ) = + ∞ .  s → 0 + s Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 9/59

lem Then, we set � x � for every ( x , t ) ∈ R N × R . A ε ( x , t ) = A ε , t , Remark Assumptions ( i ) and ( ii ) are sufficient to prove the existence of a solution of problem. The additional condition ( iii ) allows to prove the uniqueness of the solution. It has been introduced by M. Chipot (2009), in order to prove the uniqueness of the above quasilinear elliptic equation with Dirichlet boundary conditions. It implies that ω ( t ) → 0 when t → 0. In particular, if A is Lipschitz continuous in t uniformly with respect to y , i.e., if there exists L > 0 s.t. | A ( y , t ) − A ( y , t 1 ) | ≤ L | t − t 1 | , for almost every y ∈ Y and for every t , t 1 ∈ R , then the function ω ( t ) = Lt satisfies condition (iii) (as well as (i) and (ii)). Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 10/59

lem Concerning the data in the nonlinear Robin condition A ε ( x , u ε ) ∇ u ε · n + ε γ τ ε ( x ) h ( u ε ) = g ε ( x ) on Γ ε 1 , we assume that � - g is a Y -periodic function in L 2 ( ∂ S ); - τ is a positive Y -periodic function in L ∞ ( ∂ S ) � 1 and, denoted M ∂ S ( g ) = g ( y ) d σ y , we set | ∂ S | ∂ S � x � � g � x � if M ∂ S ( g ) = 0 , ε τ ε ( x ) = τ , g ε ( x ) = � x � ε ε g if M ∂ S ( g ) � = 0 . ε Remark This is the good scaling for g ε in order or to have uniform and non trivial a priori estimates. Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 11/59

lem Concerning the nonlinear term h ( u ε ) we assume that  ( i ) h is an increasing function in C 1 ( R ), with h (0) = 0;      ( ii ) there exists a constant C > 0 and an exponent q with  N  1 ≤ q < ∞ if N = 2 and 1 ≤ q ≤ N − 2 if N > 2 ,     � �  � h ′ ( s ) � ≤ C (1 + | s | q − 1 ) . such that ∀ s ∈ R , Remarks ⋆ These assumptions have been introduced by D. Cioranescu, P. D. and R. Zaki for the homogenization of the linear elliptic equation with nonlinear Robin conditions. ⋆ In particular, they imply that sh ( s ) ≥ 0 and, for some positive constant C 1 , | h ( s ) | ≤ C 1 (1 + | s | q ) ∀ s ∈ R . Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 12/59

lem Some physical motivations For several composites the thermal conductivity depends in a nonlinear way on the temperature itself (see the Encyclopedia of Material Science and Engineering). For instance, for glass or wood, where the conductivity is nonlinearly increasing with the temperature, as well as ceramics, where it is decreasing, or aluminium and semi-conductors, where the dependence is not even monotone. Nonlinear Robin conditions appear in several physical situations such as climatization (see C. Timofte, Stud. Univ. Babes-Bolyai Math., 2007) or some chemical reactions (see C. Conca, J.I. Diaz, A. Linan and C. Timofte, Electronic Journal of Differential Equations, 2004). Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Position of the problem 13/59