Remarks on the strong maximum principle.
The paper provides a new proof of a maximum principle, using methods from PDE theory, of a theorem that has been proved by Ancona with methods from potential theory. The theorem is: Theorem (Ancona, 1979): Assume \(\Omega \subset \mathbb R^N\) is an open bounded set. Let \(u \in L^1(\Omega), u \geq 0\) a.e. in \(\Omega,\) be such that \(\Delta u\) is a Radon measure on \(\Omega.\) Then there exists \(\tilde{u}:\Omega \mapsto \mathbb R\) quasicontinuous such that \(u=\tilde{u}\) a.e. in \(\Omega.\) Let \(a \in L^1(\Omega), a \geq 0\) a.e. in \(\Omega.\) If \(-\Delta u + au \geq 0 \text{ in } \Omega, \) in the following sense \( \int_E \Delta u \leq \int_E au \) for every Borel set \(E \subset \Omega,\) and if \(\tilde u =0\) on a set of positive \(H^1\)-capacity in \(\Omega,\) then \(u=0\) a.e. in \(\Omega.\) In the paper there are discussed several possibilities to weaken the conditions in this theorem in order to obtain similar conclusions.
scientific article; zbMATH DE number 2004887