Curvature estimates in dimensions 2 and 3 for the level sets of \(p\)-harmonic functions in convex rings (Q2841364)

Let \(\Omega\) be a bounded doubly connected domain in \(\mathbb R^n\) (\(n=2,3\)) whose boundary consists of two smooth components \(\Gamma_1\) and \(\Gamma_2\). Let \(u\) satisfy the equation \(\mathrm{div}(|Du|^{p-2}Du)=0\) in \(\Omega\).NEWLINENEWLINE The authors investigate the level sets of \(u\) when \(u=0\) on \(\Gamma_1\) and \(u=0\) on \(\Gamma_2\). In particular, it is proved that if \(n=3\) and \(p \geq 2\), the Gaussian curvature of the level sets attains its minimum on \(\Gamma_1 \cup \Gamma_2\).