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.


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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