We consider the Robin boundary value problem \({\mathrm {div}}\,(A\nabla u) = {\mathrm {div}}\,\varvec{f}+F\) in \(\Omega \), a \(C^1\) domain, with \((A\nabla u - \varvec{f})\cdot {\varvec{n}}+ \alpha u = g\) on \(\Gamma \), where the matrix A belongs to \(VMO ({\mathbb {R}}^3) \), and discover the uniform estimates on \(\Vert u\Vert _{W^{1,p}(\Omega )}\), with \(1< p < \infty \), independent of \(\alpha \). 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 \(\alpha \) tends to \(\infty \) (resp. 0).

我们认为Robin边值问题\（{\ mathrm {DIV}} \（A \ nabla U）= {\ mathrm {DIV}} \，\ varvec {F} + F \）在\（\欧米茄\），一个\（C ^ 1 \）域，在\（\ Gamma \）上带有\（（A \ nabla u-\ varvec {f}）\ cdot {\ varvec {n}} + \ alpha u = g \），其中矩阵A属于\（VMO（{\ mathbb {R}} ^ 3）\），并发现\（\ Vert u \ Vert _ {W ^ {1，p}（\ Omega） } \），其中\（1 <p <\ infty \）独立于\（\ alpha \）。与情况\（p = 2，\）不同这比较简单，我们在这里将其称为弱逆荷尔不等式。该估计表明，当参数\（\ alpha \）趋于\（\ infty \）（resp。0 ）时，Robin问题的解与相应空间中Dirichlet（result。Neumann）问题的解强烈收敛。