Singular elliptic problems in perforated and two-component domains

Abstract : This thesis is mainly devoted to the study of some singular elliptic problems posed in perforated domains. Denoting by Ωɛ* e domain perforated by ɛ-periodic holes of ɛ-size, we prove existence and uniqueness of the solution , for fixed ɛ, as well as homogenization and correctors results for the following singular problem :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Where homogeneous Dirichlet and nonlinear Robin conditions are prescribed on the exterior boundary Γɛ0 and on the boundary of the holles Γɛ1, respectively. The quasilinear matrix field A is elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear singular lower order ter mis the product of a continuous function ζ (singular in zero) and f whose summability depends on the growth of ζ near its singularity. The nonlinear boundary term h is a C1 increasing function, ρ and g are periodic nonnegative functions with prescribed summabilities. To investigate the asymptotic behaviour of the problem, as ɛ -> 0, we apply the Periodic Unfolding Method by D. Cioranescu-A. Damlamian-G. Griso, adapted to perforated domains by D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki. Finally, we show existence and uniqueness of a weak solution of the same equation in a two-component domain Ω = Ω1 υ Ω2 υ Γ, being Γ the interface between the connected component Ω1 and the inclusions Ω2. More precisely we consider{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Where ν1 is the unit external vector to Ω1 and λ a nonnegative real number. Here h represents the proportionality coefficient between the continuous heat flux and the jump of the solution and it is assumed to be bounded and nonnegative on Γ.
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Federica Raimondi. Singular elliptic problems in perforated and two-component domains. Analysis of PDEs [math.AP]. Normandie Université, 2018. English. ⟨NNT : 2018NORMR093⟩. ⟨tel-01995971⟩

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