On the Strong Comparison Principle for Degenerate Elliptic Problems with Convection

TL;DR

This paper investigates the conditions under which weak and strong comparison principles hold for degenerate elliptic problems with convection, providing new theoretical insights and counterexamples relevant to fluid flow in porous media.

Contribution

The paper establishes new sufficient conditions for the weak and strong comparison principles in degenerate elliptic problems with convection, including counterexamples when conditions are violated.

Findings

Weak comparison principle holds under general conditions.

Strong comparison principle holds under additional hypotheses.

Counterexamples show failure of strong principle without certain conditions.

Abstract

The weak and strong comparison principles, respectively, are investigated for quasi-linear elliptic boundary value problems with the $p$-Laplacian in one space dimension. We treat the degenerate case of $2 < p < \infty$ and allow also for the nontrivial convection velocity $b\colon [-1,1]\to \mathbb{R}$ in the underlying domain $(-1,1)$. We establish the weak comparison principle under a rather general, natural sufficient condition on the convection velocity, $b(x)$, and the reaction function, ${\varphi}(x,u)$. Furthermore, we establish also the strong comparison principle under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the strong comparison principle and discuss their applications to an interesting classical problem of fluid flow in porous medium, seepage flow of fluids in inclined bed. Our methods are based on a mixture of classical and new techniques.