A Hopf-type boundary point lemma for pairs of solutions to quasilinear equations

TL;DR

This paper establishes a Hopf boundary point lemma for the difference of two solutions to a quasilinear PDE, extending classical boundary principles to pairs of solutions under mild regularity and boundedness conditions.

Contribution

It introduces a Hopf-type boundary point lemma specifically for pairs of solutions to quasilinear equations, broadening the scope of boundary behavior analysis.

Findings

Proves a Hopf boundary point lemma for pairs of solutions

Extends classical boundary principles to quasilinear equations

Operates under mild boundedness and regularity assumptions

Abstract

We show a Hopf boundary point lemma for $u=u_{1}-u_{2},$ given $u_{1},u_{2} \in C^{1,\alpha}$ each weak solutions to a quasilinear equation $\sum_{i=1}^{n} D_{i}(A^{i}(x,u,Du))+B(x,u,Du)=0$ under mild boundedness assumptions on $A^{1},\ldots,A^{n},B.$