Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

Given a linear elliptic equation ∑ aᵢⱼuᵢⱼ=0 in R³ , 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 (aᵢⱼ) 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 (aᵢⱼ) , which is sharp by Martinez-Maure’s example: if K denotes the ratio between the largest and smallest eigenvalues of (aᵢⱼ) , we assume K|O lies in Lloc¹ for some connected open set O⊂ S² that intersects any configuration of four disjoint closed geodesic arcs of length π in S² . 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.