Parabolic Boundary Harnack Principles in Domains with Thin Lipschitz Complement (1401.7599v1)

Abstract: We prove forward and backward parabolic boundary Harnack principles for nonnegative solutions of the heat equation in the complements of thin parabolic Lipschitz sets given as subgraphs $E={(x,t): x_{n-1}\leq f(x'',t),x_n=0}\subset \mathbb{R}^{n-1}\times\mathbb{R} $ for parabolically Lipschitz functions $f$ on $\mathbb{R}^{n-2}\times\mathbb{R}$. We are motivated by applications to parabolic free boundary problems with thin (i.e co-dimension two) free boundaries. In particular, at the end of the paper we show how to prove the spatial $C^{1,\alpha}$ regularity of the free boundary in the parabolic Signorini problem.