Harnack inequality for non-uniformly elliptic equations in non-divergence form

TL;DR

This paper establishes new regularity results, including Harnack inequalities, for solutions to degenerate elliptic equations with coefficients satisfying certain integrability conditions, extending known theory.

Contribution

It proves Harnack and Weak Harnack inequalities for non-uniformly elliptic equations under minimal assumptions on coefficient degeneracy, and introduces new maximum principles.

Findings

Harnack inequality holds for p sufficiently large depending on dimension

New log-L^ε Weak Harnack inequality for supersolutions

Counterexamples show inequalities fail for p < d-1

Abstract

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We prove an improvement of oscillation and a Liouville theorem for $p>d-1$, and a Harnack inequality for $p$ sufficiently large depending on dimension. Along the way, we obtain a new $\log-L^\varepsilon$ Weak Harnack inequality for supersolutions. Then, touching subsolutions by double exponential blow-up barriers, we also derive a logarithmic local maximum principle that is new even in the uniformly elliptic case. Both of these results hold for $p>d-1$. Finally, we construct examples showing that there cannot be Harnack or Weak Harnack inequalities in the regime $p<d-1$, nor can there be power-type $L^\varepsilon$ inequalities in the case of any $p<\infty$.