Singular perturbation method for inhomogeneous nonlinear free boundary problems



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 



We would like to thank the anonymous referee for the careful reading and nice and detailed suggestions throughout the paper. The first author was partially supported by CNPq-Brazil and Funcap-Ceará. The second author was partially supported by NSF grant DMS-0701392, NSFC 11031001 and Hongshan Xiaan foundation.


  1. 1.
    Alt, H., Caffarelli, L.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)MATHMathSciNetGoogle Scholar
  2. 2.
    Berestycki, H., Caffarelli, L., Nirenberg, L.: Uniform Estimates for Regularization of Free Boundary Problems. Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., vol. 122. Dekker, New York, pp. 567–619 (1990)Google Scholar
  3. 3.
    Caffarelli, L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189–213 (1989)Google Scholar
  4. 4.
    Caffarelli, L.: A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz. Comm. Pure Appl. Math. 42(1), 55–78 (1989)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Caffarelli, L.: A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on \(X\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(4), 583–602 (1989)Google Scholar
  6. 6.
    Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence (1995)Google Scholar
  7. 7.
    Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Caffarelli, L., Jerison, D., Kenig, C.: Regularity for inhomogeneous two-phase free boundary problems. Part I: Flat free boundaries are \(C^{1,\alpha }.\) In preparationGoogle Scholar
  9. 9.
    Chipot, M.: Elliptic Equations: An Introductory Course. Birkhuser Advanced Texts: Basel Textbooks, Birkhuser Verlag, Basel (2009). ISBN: 978-3-7643-9981-8Google Scholar
  10. 10.
    Crandall, M.G., Kocan, M., Lions, P.L., Swiech, A.: Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations. Electron. J. Differ. Equ. 24, 1–22 (1999)MathSciNetGoogle Scholar
  11. 11.
    Crandall, M.G., Kocan, M., Soravia, P., Swiech, A.: On the Equivalence of Various weak Notions of Solutions of Elliptic PDES with Measurable Ingredients Pitman Res. Notes Math. Ser., 350, Longman, Harlow (1996)Google Scholar
  12. 12.
    Caffarelli, L., Lederman, C., Wolanski, N.: Uniform estimates and limits for a two phase parabolic singular perturbation problem. Indiana Univ. Math. J. 46(2), 453–489 (1997)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Caffarelli, L., Lederman, C., Wolanski, N.: Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem. Indiana Univ. Math. J. 46(3), 719–740 (1997)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Crandall, M.G.: Viscosity Solutions: A Primer. Viscosity Solutions and Applications (Montecatini Terme, 1995), pp. 1–43, Lecture Notes in Math., 1660, Springer, Berlin (1995)Google Scholar
  15. 15.
    Caffarelli, L., Salsa, S.: Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics—Volume 68—AMS (2005)Google Scholar
  16. 16.
    Caffarelli, L., Vázquez, J.L.: A free-boundary problem for the heat equation arising in flame propagation. Trans. Am. Math. Soc. 347(2), 411–441 (1995)CrossRefMATHGoogle Scholar
  17. 17.
    De Silva, D.: Free boundary regularity for a problem with right hand side. Interf. Free Bound. 13(2), 223–238 (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Danielli, D., Petrosyan, A., Shahgholian, H.: A singular perturbation problem for the \(p\)-Laplace operator. Indiana Univ. Math. J. 52(2), 457–476 (2003)MATHMathSciNetGoogle Scholar
  19. 19.
    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations. Comm. Partial Differ. Equ. 16(2–3), 311–361 (1991)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Lederman, C., Wolanski, N.: Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27(2), 253–288 (1999)Google Scholar
  21. 21.
    Lederman, C., Wolanski, N.: A two phase elliptic singular perturbation problem with a forcing term. J. Math. Pures Appl. (9) 86(6), 552–589 (2006)Google Scholar
  22. 22.
    Moreira, D.R.: Least supersolution approach to regularizing free boundary problems. Arch. Ration. Mech. Anal. 191(1), 97–141 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Moreira, D.R.: Erratum: Least supersolution approach to regularizing free boundary problems. Arch. Ration. Mech. Anal. 194(2), 713–716 (2009)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Martnez, S., Wolanski, N.: A singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of Lieberman. SIAM J. Math. Anal. 4(1), 318–359 (2009)CrossRefGoogle Scholar
  25. 25.
    Moreira, D.R., Wang, L.: Hausdorff measure estimates and Lipschitz regularity in inhomogeneous nonlinear free boundary problems, PreprintGoogle Scholar
  26. 26.
    Maly, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence (1997). ISBN: 0-8218-0335-2Google Scholar
  27. 27.
    Pucci, P., Serrin, J.: The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, 73. Birkhauser Verlag, Basel (2007). ISBN: 978-3-7643-8144-8Google Scholar
  28. 28.
    Ricarte, G., Teixeira, E.: Fully nonlinear singularly perturbed equations and asymptotic free boundaries. J. Funct. Anal. 261(6), 1624–1673 (2011)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Swiech, A.: \(W^{1, p}\)-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Differ. Equ. 2(6), 1005–1027 (1997)MATHMathSciNetGoogle Scholar
  30. 30.
    Teixeira, E.: Optimal regularity of viscosity solutions of fully nonlinear singular equations and their limiting free boundary problems. XIV School on Differential Geometry (Portuguese). Mat. Contemp. 30, 217–237 (2006)MATHMathSciNetGoogle Scholar
  31. 31.
    Trudinger, N.S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. XX, 721–747 (1967)Google Scholar

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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