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The weak maximum principle for solutions of degenerate elliptic equations with lower order terms

David Cruz-Uribe, Scott Rodney

TL;DR

This work proves a weak maximum principle for subsolutions of a degenerate elliptic operator $L=-v^{-1}\mathrm{div}(Q\nabla u)+\mathbf H\cdot \mathbf R u+\mathbf S'\mathbf G u+F u$ with lower-order terms on bounded domains, extending prior existence results by the authors and Çetin, Dal, and Zeren. It develops a matrix-weighted Sobolev framework, $QH^1_0(v,\Omega)$, and leverages degenerate Sobolev/Orlicz inequalities together with degenerate subunit vector fields to obtain maximum principle estimates. The proof avoids Moser or De Giorgi iterations by using a carefully chosen test function $\varphi=(u-r)^+$ and a product-rule lemma within this weighted setting, establishing both the $Lu\le0$ and $Lu\ge0$ cases under several gain conditions. These results broaden the theory of Dirichlet problems for degenerate elliptic PDEs with lower-order terms and provide an elementary, robust approach applicable to varying degeneracy profiles.

Abstract

We prove a weak maximum principle for subsolutions of a degenerate, linear, second order elliptic operator with lower order terms, building on the existence results recently proved by the authors and Çetin, Dal and Zeren.

The weak maximum principle for solutions of degenerate elliptic equations with lower order terms

TL;DR

This work proves a weak maximum principle for subsolutions of a degenerate elliptic operator with lower-order terms on bounded domains, extending prior existence results by the authors and Çetin, Dal, and Zeren. It develops a matrix-weighted Sobolev framework, , and leverages degenerate Sobolev/Orlicz inequalities together with degenerate subunit vector fields to obtain maximum principle estimates. The proof avoids Moser or De Giorgi iterations by using a carefully chosen test function and a product-rule lemma within this weighted setting, establishing both the and cases under several gain conditions. These results broaden the theory of Dirichlet problems for degenerate elliptic PDEs with lower-order terms and provide an elementary, robust approach applicable to varying degeneracy profiles.

Abstract

We prove a weak maximum principle for subsolutions of a degenerate, linear, second order elliptic operator with lower order terms, building on the existence results recently proved by the authors and Çetin, Dal and Zeren.

Paper Structure

This paper contains 4 sections, 3 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Solution spaces for degenerate elliptic PDEs
  3. Degenerate elliptic equations
  4. Statement and proof of the weak maximum principle

Key Result

Lemma 2.3

Let $v\in L^1(\Omega)$ and $Q$ satisfy eqn:upperellipticity. If $r>0$ and $(u,{\bf g})\in QH^1_0(v,\Omega)$, then each of are elements of $QH^1_0(v,\Omega)$.

Theorems & Definitions (10)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 3.1: Lemma 4.3 of Tubitak1
  • Definition 3.2
  • Remark 3.3
  • Remark 4.2
  • Theorem 4.3
  • proof