Uniform W^1,p estimate for elliptic operator with Robin boundary condition in C¹ domain

2018/05/24 by Chérif Amrouche, Amrouche, Cherif, Carlos Conca +5 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1805.09519

Abstract

We consider the Robin boundary value problem div (A ∇ u) = div f+F in Ω, C¹ domain, with (A ∇ u - f)⋅ n + αu = g on Γ, where the matrix A belongs to VMO (ℝ³) , and discover the uniform estimates on ‖u‖_W^1,p(Ω), with 1 < p < ∞, independent on α. At the difference with the case p = 2, which is simpler, we call here the weak reverse Hölder inequality. This estimates show that the solution of Robin problem converges strongly to the solution of Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter α tends to ∞ (resp. 0).

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• Multiplicity of solutions for semilinear Robin problems involving sign-changing nonlinearities
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