Publicacions Matemàtiques

On Viscosity Solutions to the Dirichlet Problem for Elliptic Branches of Inhomogeneous Fully Nonlinear Equations

Marco Cirant and Kevin R. Payne

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For scalar fully nonlinear partial differential equations $F(x, D^2u(x)) = 0$ with $x \in \Omega \Subset \mathbb{R}^N$, we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem, where $F(x, \cdot)$ need not be monotone on all of $\mathcal{S}(N)$, the space of symmetric $N \times N$ matrices. We treat admissible viscosity solutions $u$ of elliptic branches of the equation in the sense of Krylov [20] and extend the program initiated by Harvey and Lawson [11] in the homogeneous case when $F$ does not depend on $x$. In particular, for the set valued map $\Theta$ defining the elliptic branch by way of the differential inclusion $D^2u(x) \in \partial \Theta(x)$, we identify a uniform continuity property which ensures the validity of the comparison principle and the applicability of Perron's method for the differential inclusion on suitably convex domains, where the needed boundary convexity is characterized by $\Theta$. Structural conditions on $F$ are then derived which ensure the existence of an elliptic map $\Theta$ with the needed regularity. Concrete applications are given in which standard structural conditions on $F$ may fail and without the request of convexity conditions in the equation. Examples include perturbed Monge-Ampère equations and equations prescribing eigenvalues of the Hessian.

Article information

Source
Publ. Mat., Volume 61, Number 2 (2017), 529-575.

Dates
Received: 15 February 2016
Revised: 27 May 2016
First available in Project Euclid: 29 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.pm/1498701623

Digital Object Identifier
doi:10.5565/PUBLMAT6121708

Mathematical Reviews number (MathSciNet)
MR3677871

Zentralblatt MATH identifier
1380.35087

Subjects
Primary: 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations 35D40: Viscosity solutions 35B51: Comparison principles

Keywords
Comparison principles Perron's method Admissible viscosity solutions Elliptic branches

Citation

Cirant, Marco; Payne, Kevin R. On Viscosity Solutions to the Dirichlet Problem for Elliptic Branches of Inhomogeneous Fully Nonlinear Equations. Publ. Mat. 61 (2017), no. 2, 529--575. doi:10.5565/PUBLMAT6121708. https://projecteuclid.org/euclid.pm/1498701623


Export citation