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Uniform Regularity Close to Cross Singularities in an Unstable Free Boundary Problem

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Abstract

We introduce a new method for the analysis of singularities in the unstable problem

$$ \Delta u = -\chi_{\{u >0 \}}, $$

which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of “supercharacteristic” growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded, the level set {u = 0} consists of two C 1-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    John Andersson

  2. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Henrik Shahgholian

  3. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo-to, 153-8914, Japan

    Georg S. Weiss

Authors
  1. John Andersson
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  2. Henrik Shahgholian
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  3. Georg S. Weiss
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Corresponding author

Correspondence to Henrik Shahgholian.

Additional information

Communicated by P. Constantin

Dedicated to Nina Nikolaevna Uraltseva on the occasion of her 75th birthday

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Andersson, J., Shahgholian, H. & Weiss, G.S. Uniform Regularity Close to Cross Singularities in an Unstable Free Boundary Problem. Commun. Math. Phys. 296, 251–270 (2010). https://doi.org/10.1007/s00220-010-1015-x

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  • DOI: https://doi.org/10.1007/s00220-010-1015-x

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