A Regularity Theory for Stationary Two-valued Harmonic Functions

Abstract

For stationary two-valued harmonic functions with Hölder regularity, we establish their Lipschitz regularity and prove that the nodal set consists of analytic hypersurfaces away from a singular set. The main tools are the Almgren monotonicity formula and the blow-up method. These results are applicable to some limiting problems in segregation models.

The main result of this paper has already been proved in [Tavares-Terracini, CVPDE, 2012] and [Tavares-Terracini-Verzini, CPAM, 2010], corresponding to reference [32] here and the reference therein