Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

TL;DR

This paper establishes the conditions under which degree-one homogeneous solutions to certain degenerate elliptic equations in three dimensions must be linear, extending known results to a sharp degenerate ellipticity condition.

Contribution

The paper proves linearity of solutions under a sharp degenerate ellipticity condition involving eigenvalue ratios, resolving an open problem and refining previous theorems.

Findings

Proves linearity of solutions under specific degenerate ellipticity conditions.

Identifies sharp conditions on eigenvalue ratios for solution linearity.

Extends classical results to degenerate elliptic equations in three dimensions.

Abstract

Given a linear elliptic equation $\sum a_{ij} u_{ij} =0$ in $\mathbb{R}^3$, it is a classical problem to determine if its degree-one homogeneous solutions $u$ are linear. The answer is negative in general, by a construction of Martinez-Maure. In contrast, the answer is affirmative in the uniformly elliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a known open problem to determine the degenerate ellipticity condition on $(a_{ij})$ under which this theorem still holds. In this paper we solve this problem. We prove the linearity of $u$ under the following degenerate ellipticity condition for $(a_{ij})$, which is sharp by Martinez-Maure example: if $\mathcal{K}$ denotes the ratio between the largest and smallest eigenvalues of $(a_{ij})$, we assume $\mathcal{K}|_{\mathcal{O}}$ lies in $L_{\rm loc}^1$ for some connected open set $\mathcal{O}\subset \mathbb{S}^2$ that intersects any configuration of four disjoint closed geodesic arcs of length $\pi$ in $\mathbb{S}^2$. Our results also give the sharpest possible version under which an old conjecture by Alexandrov, Koutroufiotis and Nirenberg (disproved by Martinez-Maure's example) holds.