Bootstrap regularity for integro-differential operators and its   application to nonlocal minimal surfaces
  We prove that $C^{1,\alpha}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest. 
Bego\~na Barrios Barrera, Alessio Figalli and Enrico Valdinoci
math.AP