On a conjecture of J. Serrin

Abstract

In 1964 J. Serrin proposed the following conjecture. Let $u\in W_{\mathrm{loc}}^{1,1}(\Omega )$ be a weak solution of the second order elliptic equation (1) below, in divergence form, with Hölder continuous coefficients $a_{ij}(x)$; then $u$ is a “classical” solution. We announce a solution of this conjecture assuming only $u\in B{V}_{\mathrm{loc}}(\Omega )$ and Dini continuous coefficients.