TY - JOUR
T1 - A free boundary problem related to thermal insulation
T2 - flat implies smooth
AU - Kriventsov, Dennis
N1 - Funding Information:
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.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
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.
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U2 - 10.1007/s00526-019-1509-0
DO - 10.1007/s00526-019-1509-0
M3 - Article
AN - SCOPUS:85063463979
SN - 0944-2669
VL - 58
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 2
M1 - 78
ER -