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Improved regularity for the porous medium equation along zero level sets

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Abstract

In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets. We work under a proximity regime on the exponent governing the nonlinearity of the problem. We prove that solutions are locally of class \({{\cal C}^{1 - ,{1 \over 2} - }}\) along free boundary points x0 ∈ ∂{u > 0}. Our argument consists of importing information from the heat equation, through approximation and localization methods.

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Acknowledgements

EP is partly supported by CNPq-Brazil (433623/2018-7 and 307500/2017-9), FAPERJ (E-26/200.002/2018) and Instituto Serrapilheira (1811-25904). MS is partly funded by PUC-Rio Archimedes Fund. This work started during a visit of the authors to the International Centre for Theoretical Physics (ICTP, Trieste); the authors are grateful for the kind hospitality and vibrant scientific environment offered by ICTP.

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Authors and Affiliations

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

    Edgard A. Pimentel

  2. Department of Mathematics, Pontifical Catholic University of Rio de Janeiro - PUC-Rio, 22451-900, Gávea, Rio de Janeiro-RJ, Brazil

    Edgard A. Pimentel

  3. Centro de Investigación en Matemáticas, 36000, Guanajuato Gto., Mexico

    Makson S. Santos

Authors
  1. Edgard A. Pimentel
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  2. Makson S. Santos
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Correspondence to Edgard A. Pimentel.

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Pimentel, E.A., Santos, M.S. Improved regularity for the porous medium equation along zero level sets. Isr. J. Math. 245, 135–151 (2021). https://doi.org/10.1007/s11856-021-2208-z

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  • DOI: https://doi.org/10.1007/s11856-021-2208-z

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