Homogeneous solutions of elliptic equations: the 4 semicircles theorem

Author: Pablo Mira Abstract: An old conjecture by Alexandrov, Koutrofiotis and Nirenberg states that every 1-homogeneous solution to a linear elliptic equation in Euclidean 3-space must be linear. A striking counterexample to this claim was found by Martinez-Maure in 2001. In it, the Hessian of the solution vanishes exactly at 4 disjoint geodesic semicircles of the unit sphere, and along them the equation is not uniformly elliptic. In this talk we prove the converse of this result: for any (non-linear) homogeneous solution of a linear elliptic equation in Euclidean 3-space, there must exist four disjoint geodesic semicircles in the unit sphere along which the Hessian of u vanishes, and the uniform ellipticity of the equation is lost. The result is sharp, by Martinez-Maure's example. Joint work with Jose A. Galvez.