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Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

Abstract : In the present paper we perform the homogenization of the semilinear elliptic problem: ---------- u^ε ≥ 0 in Ω^ε, ---------- - div A(x) Du^ε = F(x, u^ε) in Ω^ε, ---------- u^ε= 0 on ∂Ω^ε. ---------- In this problem F(x, s) is a Carathéodory function such that 0 ≤ F(x, s) ≤ h(x) / Γ(s) a.e. x in Ω for every s > 0, with h in some L^r(Ω) and Γ a C^1([0, + ∞[) function such that Γ(0) = 0 and Γ'(s) > 0 for every s > 0. On the other hand the open sets Ω^ε are obtained by removing many small holes from a fixed open set Ω in such a way that a "strange term" µ u^0 appears in the limit equation in the case where the function F(x, s) depends only on x. ---------- We already treated this problem in the case of a "mild singularity", namely in the case where the function F(x, s) satisfies 0 ≤ F(x, s) ≤ h(x) (1 + 1 / s). In this case the solution u^ε to the problem belongs to H^1_0 (Ω^ε) and its definition is a "natural" and rather usual one. ---------- In the general case where F(x, s) exhibits a "strong singularity" at u = 0, which is the purpose of the present paper, the solution u^ε to the problem only belongs to H^1_loc (Ω^ε) but in general does not belongs to H^1_0 (Ω^ε) any more, even if u^ε vanishes on ∂Ω^ε in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results. ---------- In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the "strange term" µ u^0 still appears in the left-hand side while the source term F(x, u^0) is not modified in the right-hand side.
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https://hal.sorbonne-universite.fr/hal-01528064
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Submitted on : Friday, May 26, 2017 - 8:32:27 PM
Last modification on : Wednesday, December 9, 2020 - 3:05:31 PM
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Daniela Giachetti, Pedro J. Martinez-Aparicio, François Murat. Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes. 2017. ⟨hal-01528064⟩

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