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

Marco Cirant and Kevin R. Payne

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Comparison principles Perron's method Admissible viscosity solutions Elliptic branches


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.

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