Regularity of the derivatives of solutions to certain degenerate elliptic equations. (English) Zbl 0554.35048

Lewis, John L.

Indiana Univ. Math. J. 32, 849-858 (1983).

The author considers the degenerate elliptic equation \(div(| \nabla u|^{p-2}\nabla u)=0\) on some domain \(\Omega \subset {\mathbb{R}}^ n\) with \(p\in (1,\infty)\) and poves that \(u\in W^ p_ 1\) belongs in fact to \(C^{1,\alpha}_{loc}\) with some \(\alpha =\alpha (p,n)\), thereby extending results of N. Ural’tseva [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 7, 184-221 (1968; Zbl 0196.125)] and L. C. Evans [J. Differ. Equations 45, 356-373 (1982; Zbl 0508.35036), different proof] to the case \(1<p<2\). Similar results for systems of this type were achieved by P. Tolksdorf [Ann. Mat. Pura Appl., IV. Ser. 134, 241-266 (1983; Zbl 0538.35034)].

Reviewer: M.Wiegner

MSC:

35J70	Degenerate elliptic equations
35D10	Regularity of generalized solutions of PDE (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

regularity

Citations:

Zbl 0196.125; Zbl 0508.35036; Zbl 0538.35034