Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient

Abstract

We prove non-uniqueness and study the behaviour of viscosity solutions of a class of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type, with quadratic growth in the gradient. The crucial a priori bound for the solutions is proved through an argument which uses a boundary growth lemma, and consequences such as boundary "half"-Harnack inequalities, which are of independent interest. Our results are new even for linear equations.

9 pages

References in corpus (1)

• A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient

Cited by in corpus (2)

• Inhomogeneous Hopf-Oleĭnik Lemma and Applications. Part IV: Sharp Krylov Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with unbounded coefficients and $C^{1,Dini}$ boundary data
• Boundary weak Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE