Fully nonlinear parabolic fixed transmission problems

TL;DR

This paper studies fully nonlinear parabolic transmission problems across a time-dependent interface, proving regularity of solutions and establishing key estimates and well-posedness results.

Contribution

It introduces new regularity results for viscosity solutions and a novel ABP-Krylov-Tso estimate for these complex transmission problems.

Findings

Viscosity solutions are piecewise $C^{1,rac{eta}{2}}$ up to the interface.

Established existence and uniqueness of solutions.

Proved a comparison principle for the problem.

Abstract

We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is $C^{1,\alpha}$, in the parabolic sense, then viscosity solutions are piecewise $C^{1,\alpha}$ up to the interface. As byproducts, we obtain a new ABP-Krylov-Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.