Strong Maximum Principle and Boundary Estimates for Nonhomogeneous Elliptic Equations

Abstract

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the form $$ F(x,u,Du,D^{2}u) = 0 $$ F ( x , u , D u , D 2 u ) = 0 under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.

Estimates of $p$-harmonic functions in planar sectors

“…Indeed, we conclude that solutions must vanish at the same rate as |x| k(ν,p) as x approaches the apex (Corollary 5.1). Similar growth estimates where proved in the setting of C 1,1 -domains in [4] and for wider classes of equations and other geometric settings in [2], [38]- [40], [46], [47] and [49]. An immediate consequence of Corollary 5.1 is the boundary Harnack inequality for p-harmonic functions in planar sectors, see (5.3).…”