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A free boundary problem related to thermal insulation: flat implies smooth

  • Dennis KriventsovEmail author
Article
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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
$$\begin{aligned} F_1(A,u) = \int _A |\nabla u|^2 d{{\mathcal {L}}}^n + \int _{\partial A} u^2 + \bar{C} {{\mathcal {L}}}^n(A) \end{aligned}$$
over all pairs (Au) of open sets A containing a fixed set \(\Omega \) and functions \(u\in H^1(A)\) which equal 1 on \(\Omega \), the boundary \(\partial A\) locally coincides with the union of the graphs of two \(C^{1,\alpha }\) 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.

Mathematics Subject Classification

35R35 35J20 

Notes

Acknowledgements

The author is grateful for all of the help and encouragement he was given Luis Caffarelli, without whom this project would not have been possible. He was supported by NSF Grant DMS-1065926 and the NSF MSPRF fellowship DMS-1502852.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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