Branch points for (almost-)minimizers of two-phase free boundary problems

Guy David; Max Engelstein; Mariana Smit Vega Garcia; Tatiana Toro

doi:10.1017/fms.2022.105

Branch points for (almost-)minimizers of two-phase free boundary problems

Guy David, Max Engelstein, Mariana Smit Vega Garcia, Tatiana Toro

School of Mathematics

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

1 Scopus citations

Abstract

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

Original language	English (US)
Article number	e1
Journal	Forum of Mathematics, Sigma
Volume	11
DOIs	https://doi.org/10.1017/fms.2022.105
State	Published - Jan 5 2023

Bibliographical note

Funding Information:
G.D. was partially supported by the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014, the European H2020 Grant GHAIA 777822, and the Simons Collaborations in MPS Grant 601941, GD. M.E. has been partially supported by the NSF grant DMS 2000288. M.S.V.G. has been partially supported by the NSF grant DMS 2054282. T.T. was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, and by NSF grant DMS-1954545.

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© The Author(s), 2023.

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title = "Branch points for (almost-)minimizers of two-phase free boundary problems",

abstract = "We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.",

author = "Guy David and Max Engelstein and Garcia, {Mariana Smit Vega} and Tatiana Toro",

note = "Funding Information: G.D. was partially supported by the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014, the European H2020 Grant GHAIA 777822, and the Simons Collaborations in MPS Grant 601941, GD. M.E. has been partially supported by the NSF grant DMS 2000288. M.S.V.G. has been partially supported by the NSF grant DMS 2054282. T.T. was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, and by NSF grant DMS-1954545. Publisher Copyright: {\textcopyright} The Author(s), 2023.",

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AU - Engelstein, Max

AU - Garcia, Mariana Smit Vega

AU - Toro, Tatiana

N1 - Funding Information: G.D. was partially supported by the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014, the European H2020 Grant GHAIA 777822, and the Simons Collaborations in MPS Grant 601941, GD. M.E. has been partially supported by the NSF grant DMS 2000288. M.S.V.G. has been partially supported by the NSF grant DMS 2054282. T.T. was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, and by NSF grant DMS-1954545. Publisher Copyright: © The Author(s), 2023.

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Y1 - 2023/1/5

N2 - We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

AB - We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

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JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

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ER -

Branch points for (almost-)minimizers of two-phase free boundary problems

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