Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

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 1-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’s 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.