On the convexity of geometric functionals of level for solutions of certain elliptic partial differential equations (Q1822687)

If \(F(s)=\int_{\{\psi (x)=s\}}g(x) dH_{n-1}(x)\), where \(H_{n-1}\) denotes the (n-1)-dimensional Hausdorff-measure, and g and \(\psi\) are sufficiently regular with \(| \nabla \psi | >0\), one may differentiate with respect to the level sets. The author derives a formula for \(d^ nF(s)/ds^ n\) in terms of the operators \(A_ 1g=(\Delta \psi /| \nabla \psi |^ 2)g,\) \(A_ 2g=(\nabla \psi /| \nabla \psi |)\nabla (g/| \nabla \psi |)\) and \(A_ 3g=(1/| \nabla \psi |)\Delta (g/| \nabla \psi |).\) These formulas are then applied to derive some convexity results for the length of the level curves of 2D-solutions of certain elliptic equations. Also solutions of \(div(| \nabla u|^{p-2} \text{grad} u)=0\) on annular regions in \({\mathbb{R}}^ m\) with convex boundaries are studied, and an isoperimetric inequality for the p-capacity in 2D is derived.