Singular perturbation method for inhomogeneous nonlinear free boundary problems

Article

Abstract

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: \( F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) \) and \( \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)\), where \(\beta _{\varepsilon }\) approaches Dirac \(\delta _{0}\) as \(\varepsilon \rightarrow 0\) and \(f_{\varepsilon }\) has a uniform control in \(L^{q}, q>N.\) Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the \(\varepsilon -\)level surfaces are established for these variational and nonvaritional solutions. Finally, letting \(\varepsilon \rightarrow 0\) basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.

Mathematics Subject Classification (2000)

Primary 35J60 35R35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de MatemáticaUFC, Bloco 914, Campus do PiciFortalezaBrazil
  2. 2.Department of MathematicsUniversity of IowaIowa CityUSA
  3. 3.Department of MathematicsShanghai Jiaotong UniversityShanghaiChina

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