Estimates for viscosity solutions of fully nonlinear equations near smooth boundaries

TL;DR

This paper develops a reduction technique for decay estimates of viscosity solutions of fully nonlinear PDEs near smooth boundaries, enabling boundary Harnack inequalities and regularity results even with degenerate ellipticity.

Contribution

It introduces a novel reduction method that simplifies boundary decay estimates to one-dimensional inequalities, accommodating degenerate ellipticity and broad classes of nonlinear PDEs.

Findings

Derived boundary Harnack inequalities for nonlinear equations near $C^{1,1}$-boundaries.

Established Hölder continuity of solution quotients near flat boundaries.

Applied results to $p(x)$-harmonic and planar $ ext{infinity}$-harmonic functions.

Abstract

We reduce the problem of proving decay estimates for viscosity solutions of fully nonlinear PDEs to proving analogous estimates for solutions of one-dimensional ordinary differential inequalities. Our machinery allow the ellipticity to vanish near the boundary and permits general, possibly unbounded, lower-order terms. A key consequence is the derivation of boundary Harnack inequalities for a broad class of fully nonlinear, nonhomogeneous equations near $C^{1,1}$-boundaries. In combination with $C^{1,\alpha}$-estimates, we also obtain that quotients of positive vanishing solutions are H\"older continuous near $C^{1,1}$-boundaries.This result applies to a wide family of fully nonlinear uniformly elliptic PDEs; and for $p(x)$-harmonic functions and planar $\infty$-harmonic functions near locally flat boundaries. We end by deriving some Phragm\'en-Lindel\"of-type corollaries in unbounded domains.