On a conjecture of J. Serrin (Q993415)

Summary: \textit{J. Serrin} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 385--387 (1964; Zbl 0142.37601)] proposed the following conjecture. Let u \(\in W_{\text{loc}}^{1,1}(\Omega )\) be a weak solution of the second order elliptic equation \[ \sum_{i,j=1}^N \frac{\partial}{\partial x_j} \bigg(a_{ij} \frac{\partial u}{\partial x_i}\bigg)=0\quad\text{in }\Omega, \] 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 \(u\in BV_{\text{loc}}(\Omega )\) and Dini continuous coefficients.