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A singular perturbation free boundary problem for elliptic equations in divergence form

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Abstract

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem \({\textrm{div}(A(x)\nabla u) = \Gamma(x) \beta_\varepsilon(u)}\) , where A = A(x) is Holder continuous, β ɛ converges to the Dirac delta δ0. By studying some suitable level sets of u ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C 1,γ surface around \({\mathcal{H}^{N-1}}\) a.e. point on the free boundary.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, RLM 12.128, Austin, TX, 78712-1082, USA

    Diego R. Moreira

  2. Department of Mathematics, Rutgers University, Hill Center-Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Eduardo V. Teixeira

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  1. Diego R. Moreira
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  2. Eduardo V. Teixeira
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Correspondence to Diego R. Moreira.

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Moreira, D.R., Teixeira, E.V. A singular perturbation free boundary problem for elliptic equations in divergence form. Calc. Var. 29, 161–190 (2007). https://doi.org/10.1007/s00526-006-0060-y

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  • DOI: https://doi.org/10.1007/s00526-006-0060-y

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