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A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

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Abstract

In this paper we study the fully nonlinear free boundary problem

$$\left\{\begin{array}{ll}F(D^{2}u) = 1 & {\rm almost \, everywhere \, in}\, B_{1} \cap \Omega\\ |D^{2} u| \leqq K & {\rm almost \, everywhere \, in} \, B_{1} \setminus \Omega,\end{array}\right.$$

where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W 2,n solutions are locally C 1,1 inside B 1. Under the extra condition that \({\Omega \supset \{D{u} \neq 0 \}}\) and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary \({\partial \Omega \cap B_1}\).

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Authors and Affiliations

  1. Mathematics Department, The University of Texas at Austin, Austin, TX, 78712-1202, USA

    Alessio Figalli

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

    Henrik Shahgholian

Authors
  1. Alessio Figalli
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  2. Henrik Shahgholian
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Correspondence to Henrik Shahgholian.

Additional information

Communicated by F. Lin

A. Figalli was partially supported by NSF Grant DMS-0969962. H. Shahgholian was partially supported by Swedish Research Council. A. Figalli acknowledges the Göran Gustafsson Foundation for his visiting appointment to KTH.

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Figalli, A., Shahgholian, H. A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations. Arch Rational Mech Anal 213, 269–286 (2014). https://doi.org/10.1007/s00205-014-0734-0

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  • DOI: https://doi.org/10.1007/s00205-014-0734-0

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