The Two-Phase Parabolic Signorini Problem

Abstract

Abstract. We study solutions to a variational inequality that models heat control on the boundary. This problem can be thought of as the two-phase parabolic Signorini problem. Specifically, we study variational solutions to the inequalitywithout any sign restriction on the function u. The main result states that the two free boundaries (in the topology of S := ∂Ω × (0, T ))cannot touch. i.e. Γ + ∩ Γ − = ∅, therefore reducing the study of the free boundary to the parabolic Signorini problem. The separation also allows us to show the optimal regularity of the solutions.

A spiral interface with positive Alt–Caffarelli–Friedman limit at the origin

“…Further applications also include the study of segregation problems in [7]. While the most typical use of the formula is to prove the optimal regularity of solutions or flatness of the free boundary, it can also be used for other purposes, such as to show the separation of phases in free boundary problems (see [1][2][3]).…”

Optimal Regularity and the Free Boundaryin the Parabolic Signorini Problem

“…In relation to temperature control problems on the boundary, described in [DL76], we would like to mention the two recent papers by Athanasopoulos and Caffarelli [AC10], and by Allen and Shi [AS13]. Both papers deal with two-phase problems that can be viewed as generalizations of the one-phase problem (with ϕ = 0) considered in this paper.…”

“…Both papers deal with two-phase problems that can be viewed as generalizations of the one-phase problem (with ϕ = 0) considered in this paper. The paper [AS13] establishes the phenomenon of separation of phases, thereby locally reducing the study of the two-phase problem to that of one-phase. A similar phenomenon was shown earlier in the elliptic case by Allen, Lindgren, and the third named author [ALP12].…”