On Hopf's Lemma and the Strong Maximum Principle

TL;DR

The paper investigates Hopf's Lemma and the Strong Maximum Principle for a class of non-elliptic quasilinear PDEs F(u)=sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}, showing that the near-zero behavior of the g_i governs validity. A divergent divergence integral G(ξ)=∫_0^ξ g_N(ζ^2/N)/ζ dζ serves as a key sufficient condition ensuring Hopf's Lemma and SMP, with a radial subsolution construction underpinning the comparison arguments. It also derives a sharp necessary condition in a restricted setting via counterexamples when G is finite and a specific limit involving g and g' vanishes. Additionally, the authors develop non-radial subsolutions to extend SMP results to more general, non-rotationally symmetric problems. Overall, the work clarifies when Hopf's Lemma and SMP extend beyond elliptic operators, guided by the zero-behavior of the degenerating coefficients.

Abstract

In this paper we consider Hopf's Lemma and the Strong Maximum Principle for supersolutions to a class of non elliptic equations. In particular we prove a sufficient condition for the validity of Hopf's Lemma and of the Strong Maximum Principle and we give a condition which is at once necessary for the validity of Hopf's Lemma and sufficient for the validity of the Strong Maximum Principle.

Figures (4)

• Figure 1: $\Omega$ in the case $N=2$.
• Figure 2: The sets $\omega$ and $\mathcal{A}$ in the case $N=2$ and $n=3$.
• Figure 3: $\mathcal{A}$ and $\mathcal{R}(O^*,l,l_N)$ in the case $N=2$.
• Figure 4: The set $\mathcal{R}=\mathcal{R}(q^*,l,l_2)$ in the case $N=2$.

Theorems & Definitions (14)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4: Hopf's Lemma
• Theorem 1: Strong Maximum Principle
• Theorem 2
• Remark 1
• Lemma 5
• Theorem 3: The Strong Maximum Principle