Regularity results for very degenerate elliptic equations (Q386258)

In this paper, the authors consider a function \(u: \Omega \rightarrow \mathbb R\) which locally minimizes the functional NEWLINE\[NEWLINE\int_\Omega {\mathcal F}(\nabla u)+fu,NEWLINE\]NEWLINE with \({\mathcal{F}}:\mathbb R^n \rightarrow \mathbb R\) is a convex nonnegative function such that \({\mathcal{F}}\in C^2(\mathbb R^n\setminus \bar{E})\) (E is a bounded, convex set with \(0\in \mathrm{Int}(E))\) and \(f\in L^q(\Omega),\) for some \(q>n,\) where \(n\) is a positive integer and \(\Omega \) is a bounded open subset of \(\mathbb R^n.\)NEWLINENEWLINE Using an approximation argument dealing with smooth functions, the authors prove that if \(\nabla^2{\mathcal F}\) is uniformly elliptic, then for any continuous function \({\mathcal H}:\mathbb R^n \rightarrow \mathbb R\) such that \({\mathcal H}=0\) on \(E,\) \({\mathcal H}(\nabla u)\in C^0(\Omega).\)NEWLINENEWLINE In particular, they prove that if \({\mathcal{F}}\in C^1(\mathbb R^n),\) then \(\nabla {\mathcal F}(\nabla u)\in C^0(\Omega).\)NEWLINENEWLINEThe authors generalize the result to dimension \(n\) with a general convex set of degeneracy, using a different method and following some ideas of a paper by \textit{L. Wang} [J. Differ. Equations 107, No. 2, 341--350 (1994; Zbl 0792.35067)] in the case of the \(p\)-Laplacian and extend a previous result valid only in dimension 2 (see [\textit{F. Santambrogio} and \textit{V. Vespri}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 12, 3832--3841 (2010; Zbl 1202.35107)]).