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Overdetermined problems for the fractional Laplacian in exterior and annular sets

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Abstract

We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. We also study the extension of the result in bounded non-convex regions, as well as the radial symmetry of the solution when the set is assumed a priori to be rotationally symmetric.

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Authors and Affiliations

  1. Department of Mathematics, Francesco Brioschi, Politecnico di Milano, Via Bonardi 9, 20133, Milano, Italy

    Nicola Soave

  2. Department of Mathematics and Statistics, University of Western Australia, 35 Sterling Highway, WA, 6009, Crawley, Australia

    Enrico Valdinoci

Authors
  1. Nicola Soave
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  2. Enrico Valdinoci
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Corresponding author

Correspondence to Enrico Valdinoci.

Additional information

Partially supported through the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”.

Partially supported through the project ERC Starting Grant 2011 n. 277749 “Elliptic PDEs and Symmetry of Interfaces and Layers for Odd Nonlinearities - EPSILON" and and the project PRIN Grant 201274FYK7 “Aspetti variazionali e perturbativi nei problemi differenziali nonlineari”.

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Soave, N., Valdinoci, E. Overdetermined problems for the fractional Laplacian in exterior and annular sets. JAMA 137, 101–134 (2019). https://doi.org/10.1007/s11854-018-0067-2

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  • DOI: https://doi.org/10.1007/s11854-018-0067-2

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