Uniform W1,p estimates for an elliptic operator with Robin boundary condition in a C1 domain

We consider the Robin boundary value problem div ( A ∇ u ) = div f + F in Ω , a C 1 domain, with ( A ∇ u - f ) · n + α u = g on Γ , where the matrix A belongs to V M O ( R 3 ) , and discover the uniform estimates on ‖ u ‖ W 1 , p ( Ω ) , with 1 < p < ∞ , independent of α . At the difference wi...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2020, Vol.59 (2)
Main Authors: Amrouche, C., Conca, C., Ghosh, A., Ghosh, T.
Format: Article
Language:English
Subjects:
Analysis
Analysis of PDEs
Boundary conditions
Boundary value problems
Calculus of Variations and Optimal Control
Optimization
Control
Dirichlet problem
Domains
Estimates
Functional Analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
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Summary:We consider the Robin boundary value problem div ( A ∇ u ) = div f + F in Ω , a C 1 domain, with ( A ∇ u - f ) · n + α u = g on Γ , where the matrix A belongs to V M O ( R 3 ) , and discover the uniform estimates on ‖ u ‖ W 1 , p ( Ω ) , 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).
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-020-1713-y