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Removable singularities for degenerate elliptic Pucci operators. (English) Zbl 1366.35048

In this paper, the authors consider the following nonlinear degenerate elliptic problems \[ F(u, Du, D^2u)=f(x)\;\; \text{in}\;\; \Omega, \] where \(\Omega \subset \mathbb R^n\) is a bounded domain, \(Du, D^2u\) are respectively the gradient and the Hessian matrix of the real valued function \(u, f \in C(\Omega)\) and \(F: \mathbb R\times \mathbb R^n \times S^n \rightarrow \mathbb R\) satisfying some nonlinear structure assumptions. Here \(S^n\) is the set of \(n\times n\) real symmetric matrices.
The authors give first some properties concerning viscosity solutions and uniformly elliptic operators. Further, they give a priori estimates for subsolutions of such problems involving these operators and show a counterexample to the validity of the maximum principle. Finally, they establish some extentions of these results and deal with removable singularities to highly degenerate ellipticity.

MSC:

35J70 Degenerate elliptic equations
35B50 Maximum principles in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
35D40 Viscosity solutions to PDEs
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