The two-phase parabolic signorini problem

Mark Allen, Wenhui Shi

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)


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 inequality

Z ΩT (∂tu)(w − u) + ∇u∇(w − u) dx dt

+ Z S λ+(w+ − u +) + λ−(w− − u −)dHn−1 dt ≥ 0

without any sign restriction on the function u. The main result states that the two free boundaries (in the topology of S := ∂Ω × (0, T ))

Γ + = ∂{u > 0} ∩ S and Γ − = ∂{u < 0} ∩ S

cannot touch: that is, 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. 

Original languageEnglish
Pages (from-to)727-741
Number of pages15
JournalIndiana University Mathematics Journal
Issue number2
Publication statusPublished - 1 Jan 2016


  • Free boundary
  • Separation of phases
  • Signorini
  • Two-phase parabolic problem

Cite this