The two phase parabolic Signorini Problem

TL;DR

This paper analyzes the two-phase parabolic Signorini problem, demonstrating that solutions with boundary sign conditions simplify the free boundary study to the well-understood one-phase case, establishing regularity and boundary properties.

Contribution

It proves that solutions with boundary sign conditions reduce the two-phase problem to the one-phase Signorini problem, simplifying analysis and regularity results.

Findings

Solutions with boundary sign conditions always exist locally.

The free boundary analysis can be reduced to the one-phase problem.

Optimal regularity results are established for solutions.

Abstract

We study solutions to a variational equation that models heat control on the boundary. This problem can be thought of as the two phase parabolic Signorini problem. We show that when the solution has a sign on the boundary, the study of the free boundary can be reduced to the study of the free boundary in the parabolic Signorini problem. We show that locally there is always a sign on the boundary; therefore, the optimal regularity of the solutions as well as the study of the free boundary is reduced to the one phase parabolic Signorini problem.