Regularity and polar sets for supersolutions of certain degenerate elliptic equations (Q1124041)

This paper is a continuation of papers by the authors [Acta Math. 155, 153-171 (1985; Zbl 0607.35042)] and by \textit{S. Granlund} and the authors [Trans. Am. Math. Soc. 277, 43-73 (1983; Zbl 0518.30024); Ann. Acad. Sci. Fenn., Ser. A 7, 233-247 (1982; Zbl 0468.30015) and Pac. J. Math. 125, 381-395 (1986; Zbl 0633.31004)]. A weak solution \(u\in W^ 1_ n(G)\) of the Euler equation \(\nabla \cdot \nabla_ hF(x,\nabla u(x))=0\) in an open set \(G\subset R^ n\) is called F-extremal. (The conditions assumed on F can be summarized by \(F(x,h)\approx | h|^ n.)\) \(\nu: G\to R\cup \{\infty \}\) is called super-F-extremal if it is a lower semicontinuous function, satisfying: for all \(D\subset \subset G\) and all F-extremals \(u\in C(\bar D)\) the condition \(u\leq \nu\) in \(\partial D\) implies \(u\leq \nu\) in D. Concerning the regularity of super-F-extremals it is proved that they belong to loc \(W^ 1_ p(G)\) for all \(p<n\). The set C is called F-polar if there is a super-F-extremal \(\nu\) in a neighbourhood V of C such that \(\nu |_ C=\infty\) and \(\nu\) \(\not\equiv \infty\) in each component of V. The authors characterize F- polar sets and show that F-polarity is independent of F.