Abstract
In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations \({\Delta u = \beta_{\varepsilon}(u) F(\nabla u)}\) , where β ε approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried out and the limit functions are proven to be viscosity and pointwise solution (\({\mathcal{H}^{n-1}}\) almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C 1,α surface around \({\mathcal{H}^{n-1}}\) almost everywhere point.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Axler, S., Bourdon, P., Ramey, W.: Harmonic Functions Theory. Graduate texts in Mathematics No. 137, Springer, New York, 2001
Alt, H., Caffarelli, L.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)
Berestycki, H., Caffarelli, L.A., Nirenberg, L.: Uniform Estimates for Regularization of Free Boundary Problems. Analysis and partial differential equations, pp. 567–619, Lecture Notes in Pure and Appl. Math., Vol. 122. Dekker, New York, 1990
Blair, D.: Inversion Theory and Conformal Mapping Student mathematical library, No. 9, American mathematical society, Provide, RI, 2000
Caffarelli, L.A.: A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C 1,α. Rev. Mat. Iberoamericana 3(2), 139–162 (1987)
Caffarelli, L.A.: A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz. Commun. Pure Appl. Math. 42(1), 55–78 (1989)
Caffarelli, L.A.: 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 (1988/1989)
Caffarelli, L.A.: Uniform Lipschitz regularity of a singular pertubation problem. Differ. Integral Equ. 8(7), 1585–1590 (1995)
Caffarelli, L.A., Cabre, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 1995
Caffarelli, L.A., Lee, K.-A., Mellet, A.S.: Limit and homogenization for flame propagation in periodic excitable media. Arch. Ration. Mech. Anal. 172(2), 153–190 (2004)
Caffarelli, L.A., Lee, K.-A., Mellet, A.: Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation. Commun. Pure Appl. Math. 59(4), 501–525 (2006)
Caffarelli, L.A., Jerison, D., Kenig, C.: Some new monotonicity Theorems with applications to free boundary problems. Ann. Math. 155, 369–404 (2002)
Caffarelli, L.A., Lederman, C., Wolanski, N.: Uniform estimates and limits for a two phase parabolic singular pertubation problem. Indiana Univ. Math. J. 46(2), 453–490 (1997)
Caffarelli, L.A., Lederman, C., Wolanski, N.: Pointwise and viscosity solutions for the limit ofa two phase parabolic singular perturbation problem. Indiana Univ. Math. J. 46(3), 719–740 (1997)
Caffarelli, L.A., Vazques, J.L.: A free boundary problem for the heat equation arising in flame propagation. Trans. Am. Math. Soc. 347, 411–441 (1995)
Caffarelli, L.A., Salsa, S.: Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, Vol. 68. AMS, Providence, RI, 2005
Caffarelli, L.A., Crandall, M.G., Kocan, M., Swiech, A.: On Viscosity solutions of fully nonlinear equations with measurable ingridients. Commun. Pure Appl. Math. 49(4), 365–397 (1996)
Crandal, M.G., Kocan, M., Soravia, P., Swiech, A.: On the Equivalence of Various Weak Notions of Solutions of Elliptic PDES with Measurable Ingridients. Pitman Res. Notes Math. Ser., 350, Longman, Harlow, 1996
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, CRC Press, Bocaraton, FL, 1992
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)
Ishii, H.: Perron’s method for Hamilton–Jacobi Equations. Duke Math. J. 55(2), 369–384 (1987)
Lin, F., Han, Q.: Elliptic Partial Differential Equations. Courant lectures notes. AMS, Providence, RI, 1997
Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, Vol. 14, American Mathematical Society (AMS), Providence, RI, 1997
Lederman, C., Wolanski, N.: Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular pertubation problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27(2), 253–288 (1998/1999)
Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equation with discontinuos coefficients. Ann. Scuola Norm. Sup. di Pisa 17(3), 43–77 (1963)
Moreira, D.: Least Supersolution Approach to Regularizing Elliptic Free Boundary Problems, Ph.D. Dissertation, University of Texas, Austin, 2007
Moreira, D.R., Teixeira, E.V.O.: A Singular Perturbation Free Boundary Problem for Elliptic Equations in Divergence Form. To appear in Calculus of Variations and Partial Differential Equations
Trudinger, N.: On Regularity and Existence of Viscosity Solutions of Nonlinear Second Order, Elliptic Equations. Partial differential equations and the calculus of variations, Vol. II, pp. 939–957, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser Boston, Boston, MA, 1989
Trudinger, N.: On twice differentiability of viscosity solutions of nonlinear elliptic equations. Bull. Austral. Math. Soc. 39(3), 443–447 (1989)
Zeldovich, Ya.B., Frank-Kamenestki, D.A.: The theory of thermal propagation of flames. Zh. Fiz. Khim. 12, 100–105 (1938) (in Russian); English translation in “Collected Works of Ya.B. Zeldovich”, vol 1, Princeton University Press, Princeton, 1992
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Brenier
Rights and permissions
About this article
Cite this article
Moreira, D.R. Least Supersolution Approach to Regularizing Free Boundary Problems. Arch Rational Mech Anal 191, 97–141 (2009). https://doi.org/10.1007/s00205-008-0113-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-008-0113-9