## 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 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 language | English |
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Pages (from-to) | 727-741 |

Number of pages | 15 |

Journal | Indiana University Mathematics Journal |

Volume | 65 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

## Keywords

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