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The obstacle problem for parabolic minimizers

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Abstract

We prove an existence result for parabolic minimizers of convex variational functionals with p-growth and irregular obstacles. In particular, the obstacle might be unbounded, discontinuous and satisfy no regularity assumption with respect to the time variable. Moreover, we treat the case of obstacles which do not coincide along the parabolic boundary with the prescribed time-dependent Dirichlet boundary values. The existence result for sufficiently regular obstacles coinciding on the lateral boundary with the given Dirichlet boundary values is obtained via the method of minimizing movements. More general boundary values and obstacles are treated by approximation with regular boundary values and obstacles in the sense of a stability result of solutions with respect to boundary values and the obstacles.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020, Salzburg, Austria

    Verena Bögelein

  2. Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany

    Frank Duzaar

  3. Fakultät für Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany

    Christoph Scheven

Authors
  1. Verena Bögelein
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  2. Frank Duzaar
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  3. Christoph Scheven
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Correspondence to Christoph Scheven.

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Bögelein, V., Duzaar, F. & Scheven, C. The obstacle problem for parabolic minimizers. J. Evol. Equ. 17, 1273–1310 (2017). https://doi.org/10.1007/s00028-017-0384-4

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