Uniform $W^{1,p}$ estimate for elliptic operator with Robin boundary condition in $\mathcal{C}^1$ domain

TL;DR

This paper establishes uniform W^{1,p} estimates for solutions to elliptic PDEs with Robin boundary conditions in C^1 domains, showing convergence to Dirichlet or Neumann solutions as the boundary parameter varies.

Contribution

It provides the first uniform W^{1,p} estimates for Robin problems with VMO coefficients, extending known results from the L^2 case to general p.

Findings

Uniform W^{1,p} estimates independent of boundary parameter lpha

Strong convergence of Robin solutions to Dirichlet or Neumann solutions as lpha tends to  or 0

Introduction of weak reverse Hlder inequality for p 

Abstract

We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $\Omega$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + \alpha u = g$ on $\Gamma$, where the matrix $A$ belongs to $VMO (\mathbb{R}^3) $, and discover the uniform estimates on $\|u\|_{W^{1,p}(\Omega)}$, with $1 < p < \infty$, independent on $\alpha$. At the difference with the case $p = 2,$ which is simpler, we call here the weak reverse H\"older 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 $\alpha$ tends to $\infty$ (resp. $0$).