A note on the Rellich formula in Lipschitz domains (Q1277133)

Let \(L\) be a symmetric second-order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain \(\Omega\) of \(\mathbb{R}^N\) and having Lipschitz coefficients in \(\Omega\). The so-called Rellich formula is known to hold for functions \(u\in H^2(\Omega)\). It is shown by the author that this result extends to all functions in the domain of \(L\), that is, for all \(u\in H_0^1(\Omega)\) such that \(L(u)\in L^2(\Omega)\); this amounts to a continuity property of the gradient of \(L\)-solutions with respect to perturbations of the set \(\Omega\). The proof uses well-known results and methods of the potential theory in Lipschitz domains.