Two theorems of N. Wiener for solutions of quasilinear elliptic equations. (English) Zbl 0607.35042

Lindqvist, P.; Martio, O.

Acta Math. 155, 153-171 (1985).

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 W. P. Ziemer [Arch. Ration. Mech. Anal. 92, 371-382 (1986)].

Reviewer: U.Wiegner

MSC:

35J67	Boundary values of solutions to elliptic equations and elliptic systems
35J60	Nonlinear elliptic equations
35A25	Other special methods applied to PDEs

Keywords:

capacity methods; boundary behaviour; weak solutions; Wiener’s criterion; Perron-solutions; regularity; boundary point

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