On the geometry of level sets of positive solutions of semilinear elliptic equations
Let \(\Omega \subset {\mathbb{R}}^ n\), \(n\in {\mathbb{N}}\), \(n\geq 2\), be a bounded domain with \(C^{2,\alpha}\)-smooth boundary \(\partial \Omega\), \(\alpha\in]0,1[\), such that the origin \(0\in {\mathbb{R}}^ n\) is contained in \(\Omega\). The authors study the geometry of level sets of positive classical solutions u to the Dirichlet problem \[ (D)\quad \Delta u+f(u)=0\text{ in } \Omega \text{ with } f\in C^ 1({\mathbb{R}});\quad u=0\text{ on } \partial \Omega, \] when \(\Omega\) satisfies some symmetry conditions. The main result of the present paper states that the level sets \(\Omega_ c:=\{x\in \Omega |\) \(u(x)>c\}\), \(c\in {\mathbb{R}}^+_ 0\), of any solution \(u\in C^ 2({\bar \Omega})\) to (D) with \(u(x)>0\), \(x\in \Omega\), are starshaped with respect to 0. The proof of this hinges upon well-known properties of Coxeter-groups of transformations acting in \({\mathbb{R}}^ n\) and a careful analysis of the (not necessarily finite) group of transformations in \({\mathbb{R}}^ n\) generated by the set of reflections through hyperplanes with respect to which \(\Omega\) is symmetric. This together with a crucial estimate from \textit{B. Gidas}, \textit{W. M. Ni} and \textit{L. Nirenberg} [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)] yields the inequality \(<x,\nabla u(x)>\leq 0\), \(x\in \Omega\), from which the starshapedness (w.r.t. 0) of the level sets \(\Omega_ c\) is immediate. Essentially the same result was obtained by Kawohl for a more special case [see \textit{B. Kawohl}, Appl. Anal. 16, 205-228 (1983; Zbl 0507.35035)]. As the authors point out due to the starshapedness of the level sets \(\Omega_ c\) it is also possible to prove lower a priori bounds on the \(C^ 0(\pi)\)-norm of positive smooth solutions to (D) in exactly the same manner as by \textit{Chr. Cosner} and \textit{K. Schmitt} [Proc. Am. math. Soc. 95, 47-50 (1985; Zbl 0577.35041)] where a somewhat more restrictive symmetry condition is imposed on \(\Omega\).
scientific article; zbMATH DE number 4159195