UniformW1,p estimates for an elliptic operator with Robin boundary condition in a C1 domain

Chérif Amrouche (1) , C. Conca , A. Ghosh , T. Ghosh

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We consider the Robin boundary value problem div(A∇u) = div f + F in , a C 1 domain, with (A∇u − f) • n + αu = g on , where the matrix A belongs to V M O(R 3), and discover the uniform estimates on u W 1, p () , with 1 < p < ∞, independent of 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 the Robin problem converges strongly to the solution of the Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter α tends to ∞ (resp. 0).

Domains: Mathematics [math]; Analysis of PDEs [math.AP]; Functional Analysis [math.FA]

Chérif Amrouche, C. Conca, A. Ghosh, T. Ghosh. UniformW1,p estimates for an elliptic operator with Robin boundary condition in a C1 domain. Calculus of Variations and Partial Differential Equations, 2020, 59 (2), ⟨10.1007/s00526-020-1713-y⟩. ⟨hal-03089990⟩