Free boundary regularity for almost-minimizers

Guy David; Max Engelstein; Tatiana Toro

doi:10.1016/j.aim.2019.04.059

Free boundary regularity for almost-minimizers

Guy David, Max Engelstein, Tatiana Toro

School of Mathematics

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

In this paper we study the free boundary regularity for almost-minimizers of the functional J(u)=∫Ω|∇u(x)| ² +q ₊ ² (x)χ _{u>0} (x)+q ₋ ² (x)χ _{u<0} (x)dx where q _± ∈L ^∞ (Ω). Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [4], [5], [9], [37]). Nevertheless, using a novel argument which brings together tools from potential theory and geometric measure theory, we succeed in proving that, under a non-degeneracy assumption on q _± , the free boundary is uniformly rectifiable. Furthermore, when q ₋ ≡0, and q ₊ is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a C ^1,α function (thus extending the results of [4] to almost-minimizers).

Original language	English (US)
Pages (from-to)	1109-1192
Number of pages	84
Journal	Advances in Mathematics
Volume	350
DOIs	https://doi.org/10.1016/j.aim.2019.04.059
State	Published - Jul 9 2019

Bibliographical note

Funding Information:
G. David was partially supported by the Institut Universitaire de France, the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014 and the Simons Collaborations in MPS Grant 601941 GD. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082, the University of Chicago RTG grant DMS 1246999, a NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant NSF DMS 1500771. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823, by the Robert R. & Elaine F. Phelps Professorship in Mathematics and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.☆ G. David was partially supported by the Institut Universitaire de France, the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014 and the Simons Collaborations in MPS Grant 601941 GD. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082, the University of Chicago RTG grant DMS 1246999, a NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant NSF DMS 1500771. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823, by the Robert R. & Elaine F. Phelps Professorship in Mathematics and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

Publisher Copyright:
© 2019

Keywords

Almost-minimizer
Free boundary problem
Uniform rectifiability

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10.1016/j.aim.2019.04.059

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title = "Free boundary regularity for almost-minimizers",

abstract = " In this paper we study the free boundary regularity for almost-minimizers of the functional J(u)=∫Ω|∇u(x)| 2 +q + 2 (x)χ {u>0} (x)+q − 2 (x)χ {u<0} (x)dx where q ± ∈L ∞ (Ω). Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [4], [5], [9], [37]). Nevertheless, using a novel argument which brings together tools from potential theory and geometric measure theory, we succeed in proving that, under a non-degeneracy assumption on q ± , the free boundary is uniformly rectifiable. Furthermore, when q − ≡0, and q + is H{\"o}lder continuous we show that the free boundary is almost-everywhere given as the graph of a C 1,α function (thus extending the results of [4] to almost-minimizers).",

keywords = "Almost-minimizer, Free boundary problem, Uniform rectifiability",

author = "Guy David and Max Engelstein and Tatiana Toro",

note = "Funding Information: G. David was partially supported by the Institut Universitaire de France, the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014 and the Simons Collaborations in MPS Grant 601941 GD. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082, the University of Chicago RTG grant DMS 1246999, a NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant NSF DMS 1500771. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823, by the Robert R. & Elaine F. Phelps Professorship in Mathematics and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.☆ G. David was partially supported by the Institut Universitaire de France, the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014 and the Simons Collaborations in MPS Grant 601941 GD. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082, the University of Chicago RTG grant DMS 1246999, a NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant NSF DMS 1500771. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823, by the Robert R. & Elaine F. Phelps Professorship in Mathematics and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. Publisher Copyright: {\textcopyright} 2019",

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PY - 2019/7/9

Y1 - 2019/7/9

N2 - In this paper we study the free boundary regularity for almost-minimizers of the functional J(u)=∫Ω|∇u(x)| 2 +q + 2 (x)χ {u>0} (x)+q − 2 (x)χ {u<0} (x)dx where q ± ∈L ∞ (Ω). Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [4], [5], [9], [37]). Nevertheless, using a novel argument which brings together tools from potential theory and geometric measure theory, we succeed in proving that, under a non-degeneracy assumption on q ± , the free boundary is uniformly rectifiable. Furthermore, when q − ≡0, and q + is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a C 1,α function (thus extending the results of [4] to almost-minimizers).

AB - In this paper we study the free boundary regularity for almost-minimizers of the functional J(u)=∫Ω|∇u(x)| 2 +q + 2 (x)χ {u>0} (x)+q − 2 (x)χ {u<0} (x)dx where q ± ∈L ∞ (Ω). Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [4], [5], [9], [37]). Nevertheless, using a novel argument which brings together tools from potential theory and geometric measure theory, we succeed in proving that, under a non-degeneracy assumption on q ± , the free boundary is uniformly rectifiable. Furthermore, when q − ≡0, and q + is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a C 1,α function (thus extending the results of [4] to almost-minimizers).

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Free boundary regularity for almost-minimizers

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