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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 ofMartinez-Maure. In contrast, the answer is affirmative in the uniformlyelliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a knownopen problem to determine the degenerate ellipticity condition on $(a_{ij})$under which this theorem still holds. In this paper we solve this problem. Weprove the linearity of $u$ under the following degenerate ellipticity conditionfor $(a_{ij})$, 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 someconnected open set $\mathcal{O}\subset \mathbb{S}^2$ that intersects anyconfiguration of four disjoint closed geodesic arcs of length $\pi$ in$\mathbb{S}^2$. Our results also give the sharpest possible version under whichan old conjecture by Alexandrov, Koutroufiotis and Nirenberg (disproved byMartinez-Maure's example) holds.

Linearity of homogeneous solutions to degenerate elliptic equations in dimension three