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

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. $0$).