Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

  • José A. Gálvez

    Universidad de Granada, Granada, Spain
  • Pablo Mira

    Universidad Politécnica de Cartagena, Cartagena, Spain
Linearity of homogeneous solutions to degenerate elliptic equations in dimension three cover
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Abstract

Given a linear elliptic equation in , it is a classical problem to determine if its 1-homogeneous solutions 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 under which this theorem still holds. In this paper we solve this problem. We prove the linearity of under the following degenerate ellipticity condition for , which is sharp by Martinez-Maure’s example: if denotes the ratio between the largest and smallest eigenvalues of , we assume lies in for some connected open set that intersects any configuration of four disjoint closed geodesic arcs of length in . 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.

Cite this article

José A. Gálvez, Pablo Mira, Linearity of homogeneous solutions to degenerate elliptic equations in dimension three. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1512