### Abstract

We study the regularity of the interface for a new free boundary problem introduced in Caffarelli and Kriventsov (A free boundary problem related to thermal insulation, 2015). We show that for minimizers of the functional F1(A,u)=∫A|∇u|2dLn+∫∂Au2+C¯Ln(A)over all pairs (A, u) of open sets A containing a fixed set Ω and functions u∈ H
^{1}
(A) which equal 1 on Ω , the boundary ∂A locally coincides with the union of the graphs of two C
^{1}
^{,}
^{α}
functions near most points. Specifically, this happens at all points where the interface is trapped between two planes which are sufficiently close together. The proof combines ideas introduced by Ambrosio, Fusco, and Pallara for the Mumford–Shah functional with new arguments specific to the problem considered.

Original language | English (US) |
---|---|

Article number | 78 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 58 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2019 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

}

**A free boundary problem related to thermal insulation : flat implies smooth.** / Kriventsov, Denis.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A free boundary problem related to thermal insulation

T2 - flat implies smooth

AU - Kriventsov, Denis

PY - 2019/4/1

Y1 - 2019/4/1

N2 - We study the regularity of the interface for a new free boundary problem introduced in Caffarelli and Kriventsov (A free boundary problem related to thermal insulation, 2015). We show that for minimizers of the functional F1(A,u)=∫A|∇u|2dLn+∫∂Au2+C¯Ln(A)over all pairs (A, u) of open sets A containing a fixed set Ω and functions u∈ H 1 (A) which equal 1 on Ω , the boundary ∂A locally coincides with the union of the graphs of two C 1 , α functions near most points. Specifically, this happens at all points where the interface is trapped between two planes which are sufficiently close together. The proof combines ideas introduced by Ambrosio, Fusco, and Pallara for the Mumford–Shah functional with new arguments specific to the problem considered.

AB - We study the regularity of the interface for a new free boundary problem introduced in Caffarelli and Kriventsov (A free boundary problem related to thermal insulation, 2015). We show that for minimizers of the functional F1(A,u)=∫A|∇u|2dLn+∫∂Au2+C¯Ln(A)over all pairs (A, u) of open sets A containing a fixed set Ω and functions u∈ H 1 (A) which equal 1 on Ω , the boundary ∂A locally coincides with the union of the graphs of two C 1 , α functions near most points. Specifically, this happens at all points where the interface is trapped between two planes which are sufficiently close together. The proof combines ideas introduced by Ambrosio, Fusco, and Pallara for the Mumford–Shah functional with new arguments specific to the problem considered.

UR - http://www.scopus.com/inward/record.url?scp=85063463979&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063463979&partnerID=8YFLogxK

U2 - 10.1007/s00526-019-1509-0

DO - 10.1007/s00526-019-1509-0

M3 - Article

VL - 58

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

M1 - 78

ER -