On viscosity solutions to the Dirichlet problem for elliptic branches of nonhomogeneous fully nonlinear equations

TL;DR

This paper develops a general theory for viscosity solutions of elliptic branches of fully nonlinear PDEs, establishing comparison principles and existence results for the Dirichlet problem under broad conditions.

Contribution

It introduces a framework using elliptic maps and differential inclusions to analyze viscosity solutions for non-monotone nonlinear PDEs, extending classical methods.

Findings

Comparison principle holds for weak solutions under uniform upper semicontinuity.

Unique continuous solutions exist for the Dirichlet problem with suitably convex boundary.

Applicable to non-totally degenerate equations and eigenvalue-based PDEs.

Abstract

For scalar fully nonlinear partial differential equations depending on the Hessian andspatial coordinates, we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem with continuous boundary data. In particular, we treat admissible viscosity solutions of elliptic branches of the equation, where the nonlinearity need not be monotone on all of the space of symmetric N by N matrices. An elliptic branch (in the sense of Krylov, 1995) of the equation is encoded by a set valued map from the coordinate domain into the elliptic subsets of the symmetric matrices (an elliptic map). The nonlinearity will be monotone along this map and the degenerate elliptic PDE is replaced by the a differential inclusion. Weak solutions to such differential inclusions are defined by using the notion given by Harvey-Lawson (2009) in a pointwise manner. If the elliptic map is uniformly upper semicontinuous, we show that the comparison principle holds for these weak solutions and that Perron's method yields a unique continuous solution to the associated abstract Dirichlet problem provided that the boundary is suitably convex with respect to the elliptic mapand its dual in the sense of Harvey and Lawson. When the map encodes an elliptic branch of a given PDE, these soluitions are shown to be admissible viscosity solutions of the PDE problem. Various applications are described in terms of structural conditions which ensure the existence of the needed elliptic map. Examples include non-totally degenerate equations and equations involving the eigenvalues of the Hessian and their perturbations. In certain situations, the methods employed here will be shown to operate freely, while classical viscosity approaches may not.