Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analysis of a free boundary at contact points with Lipschitz data

Authors: A. L. Karakhanyan and H. Shahgholian
Journal: Trans. Amer. Math. Soc. 367 (2015), 5141-5175
MSC (2010): Primary 35R35
Published electronically: March 4, 2015
MathSciNet review: 3335413
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider a minimization problem for the functional

$\displaystyle J(u)=\int _{B_1^+}\vert\nabla u\vert\sp 2+\lambda _{+}^2\chi _{\{u>0\}}+\lambda _{-}^2\chi _{\{u\leq 0\}} $

in the upper half ball $ B_1^+\subset \mathbb{R}^n, n\geq 2$, subject to a Lipschitz continuous Dirichlet data on $ \partial B_1^+$. More precisely we assume that $ 0\in \partial \{u>0\}$ and the derivative of the boundary data has a jump discontinuity. If $ 0\in \overline {\partial ( \{u>0\} \cap B_1^+)}$, then (for $ n=2$ or $ n\geq 3$ and the one-phase case) we prove, among other things, that the free boundary $ \partial \{u>0\}$ approaches the origin along one of the two possible planes given by

$\displaystyle \gamma x_1 = \pm x_2, $

where $ \gamma $ is an explicit constant given by the boundary data and $ \lambda _\pm $ the constants seen in the definition of $ J(u)$. Moreover the speed of the approach to $ \gamma x_1=x_2$ is uniform.

References [Enhancements On Off] (What's this?)

  • [AC] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144. MR 618549 (83a:49011)
  • [ACF] Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), no. 2, 431-461. MR 732100 (85h:49014),
  • [AG] Hans Wilhelm Alt and Gianni Gilardi, The behavior of the free boundary for the dam problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), no. 4, 571-626. MR 693780 (85c:35089a)
  • [AS] John Andersson and Henrik Shahgholian, Global solutions of the obstacle problem in half-spaces, and their impact on local stability, Calc. Var. Partial Differential Equations 23 (2005), no. 3, 271-279. MR 2142064 (2006b:35352),
  • [BZ] Garrett Birkhoff and E. H. Zarantonello, Jets, wakes, and cavities, Academic Press Inc., Publishers, New York, 1957. MR 0088230 (19,486f)
  • [CKS] Luis A. Caffarelli, Lavi Karp, and Henrik Shahgholian, Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. (2) 151 (2000), no. 1, 269-292. MR 1745013 (2001a:35188),
  • [CS] Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68, American Mathematical Society, Providence, RI, 2005. MR 2145284 (2006k:35310)
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364 (2001k:35004)
  • [Gu] Alex Gurevich, Boundary regularity for free boundary problems, Comm. Pure Appl. Math. 52 (1999), no. 3, 363-403. MR 1656068 (99m:35272),$ \langle $363::AID-CPA3$ \rangle $3.3.CO;2-L
  • [KKS] A. L. Karakhanyan, C. E. Kenig, and H. Shahgholian, The behavior of the free boundary near the fixed boundary for a minimization problem, Calc. Var. Partial Differential Equations 28 (2007), no. 1, 15-31. MR 2267752 (2008d:35240),
  • [K] Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994. MR 1282720 (96a:35040)
  • [W1] Georg Sebastian Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 (1999), no. 2, 317-326. MR 1759450 (2001b:49053),
  • [W2] Georg S. Weiss, Boundary monotonicity formulae and applications to free boundary problems. I. The elliptic case, Electron. J. Differential Equations (2004), No. 44, 12 pp. (electronic). MR 2047400 (2004m:35286)
  • [Z] Tullio Zolezzi, On weak convergence in $ L^{\infty }$, Indiana Univ. Math. J. 23 (1973/74), 765-766. MR 0328576 (48 #6918)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35R35

Retrieve articles in all journals with MSC (2010): 35R35

Additional Information

A. L. Karakhanyan
Affiliation: Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, EH9 3JZ, Edinburgh, Scotland

H. Shahgholian
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Keywords: Free boundary problem, regularity, contact points
Received by editor(s): May 22, 2012
Received by editor(s) in revised form: May 15, 2013
Published electronically: March 4, 2015
Additional Notes: The second author was partially supported by the Swedish Research Council. The authors also thank Professor Carlos Kenig for several valuable comments. The first author thanks the Göran Gustafsson Foundation for visiting appointments to KTH
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society