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Journal Articles Mathematics in Engineering Year : 2021

The ergodic limit for weak solutions of elliptic equations with Neumann boundary condition

François Murat
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Alessio Porretta
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Abstract

We consider the so-called ergodic problem for weak solutions of elliptic equations in divergence form, complemented with Neumann boundary conditions. The simplest example reads as the following boundary value problem in a bounded domain of R^N : ------ −div(A(x)∇u) + λ = H(x, ∇u) in Ω, ------ A(x)∇u · n = 0 on ∂Ω, ------ where A(x) is a coercive matrix with bounded coefficients, and H(x, ∇u) has Lipschitz growth in the gradient and measurable x-dependence with suitable growth in some Lebesgue space (typically, |H(x, ∇u)| ≤ b(x)|∇u| + f (x) for functions b(x) ∈ L^N (Ω) and f (x) ∈ L^m(Ω), m ≥ 1). We prove that there exists a unique real value λ for which the problem is solvable in Sobolev spaces and the solution is unique up to addition of a constant. We also characterize the ergodic limit, say the singular limit obtained by adding a vanishing zero order term in the equation. Our results extend to weak solutions and to data in Lebesgue spaces L^m(Ω) (or in the dual space (H1(Ω))′) previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations. ------ This paper is part of the volume "Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta", Guest Editor: Fabiana Leoni, Mathematics in Engineering, 3 (4), (2021), pp. 1-20, doi : 0.3934/mine.2021031

Dates and versions

hal-03878599 , version 1 (29-11-2022)

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François Murat, Alessio Porretta. The ergodic limit for weak solutions of elliptic equations with Neumann boundary condition. Mathematics in Engineering, 2021, 3 (4), pp.1-20. ⟨10.3934/mine.2021031⟩. ⟨hal-03878599⟩
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