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On a two-phase free boundary problem ruled by the infinity Laplacian

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Abstract

In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in B1 are universally Lipschitz continuous in B1/2, which is the optimal regularity for the problem. We make a new use of the Ishii–Lions’ method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.

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Acknowledgments

DJA is supported by CNPq grant 427070/2016-3 and grant 2019/0014 from Paraíba State Research Foundation (FAPESQ).

JMU is partially supported by FCT–FundaçãoparaaCiência e a Tecnologia, I.P., through grant SFRH/BSAB/150308/2019 and projects PTDC/MAT-PUR/28686/2017 and UTAP-EXPL/MAT/0017/2017, and by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

DJA and JMU thank the University of Central Florida for their hospitality, and DJA thanks the Abdus Salam International Centre for Theoretical Physics, where parts of this work were conducted.

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Authors and Affiliations

  1. Department of Mathematics, Universidade Federal da Paraíba, 58059-900, João Pessoa, PB, Brazil

    Damião J. Araújo & José Miguel Urbano

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Eduardo V. Teixeira

  3. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    José Miguel Urbano

Authors
  1. Damião J. Araújo
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  2. Eduardo V. Teixeira
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  3. José Miguel Urbano
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Correspondence to José Miguel Urbano.

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Araújo, D.J., Teixeira, E.V. & Urbano, J.M. On a two-phase free boundary problem ruled by the infinity Laplacian. Isr. J. Math. 245, 773–785 (2021). https://doi.org/10.1007/s11856-021-2227-9

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  • DOI: https://doi.org/10.1007/s11856-021-2227-9

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