An elliptic problem with a layer (Q2735885)

The author investigates the local regularity of a function \(u(x)\) satisfying the integral identity NEWLINE\[NEWLINE\int_\Omega Du\,Dv\,dx+ \int_\Sigma D'u\, d'v\, dx'= 0, \quad \Sigma= \Omega\cap \{x_N= 0\}NEWLINE\]NEWLINE for arbitrary function \(v(x)\) such that \(u,v\) are from \(H_{\text{loc}}^1 (\Omega)\) with traces in \(H_{\text{loc}}^1 (\Sigma)\) and with \(\text{supp\,}v \subset \Omega\). Here \(\Omega\) is an open set in \(\mathbb{R}^N\), \(D'= (\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_{N-1}})\). For nonnegative solution of considered problem the Harnack inequality is proved by using the modification of the Moser iteration method. From this result, the local Hölder continuity of solutions follows immediately.