Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates
The paper is concerned with the \(p\)-capacitary problem: \[ \text{div}(|Du|^{p-2}Du)=0\text{ in }\Omega =\mathbb{R}^{n}\backslash \overline{\Omega }_{1},\quad u=1\text{ on }\partial \Omega _{1},\;u\rightarrow 0\text{ as } |x|\rightarrow \infty ,\tag{1} \] where \(1<p<n\) and \(\Omega _{1}\) is a connected, bounded open set, starlike with respect to the origin which is assumed to belong to \(\Omega _{1}\). For the weak solutions of (1) three main properties of the function \[ P(u,x)=\frac{|Du(x)|^{p}}{u(x)^{\frac{p(n-1)}{n-p}}} \] are studied: \(i)\) a strong maximum principle; \(ii)\) the maximum value of \(P\) in the spherically symmetric configuration of \((1)\), and \(iii)\) a rescaling property in the sense that \[ P(r^{\frac{n-p}{p-1}}u(rx),x)=P(u,rx). \] As a consequence, it is given a new proof of a recent result of \textit{W. Reichel} [Z. Anal. Anwend. 15, 619-635 (1996; Zbl 0857.35010)], concerning the spherical symmetry in (1) with the overdetermined boundary condition \(|Du|=c>0\) on \(\partial \Omega _{1}\). In order to prove some of the main results the authors exploited besides two Serrin's results: \textit{J. Serrin} [Acta Math. 111, 247-302 (1964; Zbl 0128.09101)], and \textit{J. Serrin} [Acta Math. 113, 219-240 (1965; Zbl 0173.39202)], the local regularity theory for uniformly elliptic operators used from the book of \textit{O. Ladyzhenskaya} and \textit{N. Ural'tseva} [Linear and quasilinear elliptic equations (Mathematics in Science and Engineering 46, New York-London: Academic Press. XVIII) (1968; Zbl 0164.13002)], and A. D. Alexandrov's well known theorem on the characterisation of spheres.
scientific article; zbMATH DE number 1493592