A sharp minimum principle for the problem of torsionally rigidity (Q1289051)

Let \(\Omega\) be a smooth strictly convex plane domain. Let \(u\) be the solution to the equation \(\Delta u=2\) in \(\Omega\) with \(u=0\) on \(\partial\Omega\). The author proves that the function \(P(x)=| Du| ^2-2u\) attains its minimum value on \(\partial\Omega\). This result was known for \(P^\alpha(x)=| Du| ^2-2\alpha u\) with \(\alpha>1\) [\textit{G. A. Philippin}, J. Math. Anal. Appl. 68, 526-535 (1979; Zbl 0412.73035)]. The proof is carried out by using the analyticity of \(u\) in \(\Omega\). As an application, a lower bound for the minimum of \(| Du| \) in terms of the maximum value of the curvature on \(\partial\Omega\) is obtained.