Two theorems of N. Wiener for solutions of quasilinear elliptic equations (Q1085337)

The authors study by capacity methods the boundary behaviour of weak solutions \(u\in C(G)\cap H^ 1_{n,loc}(G)\) of \(div(\partial F/\partial p(x,\nabla u))=0\) on some bounded domain \(G\subset {\mathbb{R}}^ n\), where the kernel F(x,p) is conformally invariant \((F(x,\lambda p)=| \lambda |^ nF(x,p))\), strictly convex with respect to p and fulfills some regularity assumptions. First they prove the converse of Wiener's criterion, namely if \(x_ 0\in \partial G\) is regular (meaning that continuous boundary values are attained continuously by the upper and lower Perron-solutions), then \({\mathbb{R}}^ n\setminus G\) is not thin at \(x_ 0\), which means \(\int^{1}_{0}r^{-1}\phi (r)^{1/(n- 1)}dr=\infty\), where \(\phi\) (r) denotes the n-capacity of \(R^ n\setminus G\cap \bar B(x_ 0,r)\) with respect to \(B(x_ 0,2r)\). This implies that the regularity of a boundary point does not depend on F. Second they show that upper and lower Perron-solutions coincide for continuous boundary values and form a solution. For quasiminima in the case \(F\sim | p|^ q\) with \(q<n\), see \textit{W. P. Ziemer} [Arch. Ration. Mech. Anal. 92, 371-382 (1986)].