Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

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.