An eigenvalue problem for fully nonlinear elliptic equations with gradient constraints

TL;DR

This paper investigates an eigenvalue problem for a class of fully nonlinear elliptic PDEs with gradient constraints, establishing existence, uniqueness of the eigenvalue, and properties of solutions, with implications for singular ergodic control.

Contribution

It introduces a new eigenvalue problem framework for nonlinear PDEs with gradient constraints, proving existence, uniqueness, and regularity results for solutions.

Findings

Existence of a unique eigenvalue  for solutions with specified growth.

Convex solutions with bounded second derivatives under uniform ellipticity and convexity.

Uniqueness of solutions up to an additive constant in certain symmetric cases.

Abstract

We consider the problem of finding $\lambda\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{\lambda + F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic, positively homogeneous and superadditive, $f$ is convex and superlinear, and $H$ is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique $\lambda^*$ for which the above equation has a solution $u$ with appropriate growth as $|x|\rightarrow \infty$. Moreover, associated to $\lambda^*$ is a convex solution $u^*$ that has bounded second derivatives, provided $F$ is uniformly elliptic and $H$ is uniformly convex. It is unknown whether or not $u^*$ is unique up to an additive constant; however, we verify this is the case when $n=1$ or when $F, f,H$ are "rotational."