On the full regularity of the free boundary in a class of variational problems
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Abstract:
We consider nonnegative minimizers of the functional \[ J_p(u;\Omega )=\int _\Omega |\nabla u|^p+ \lambda _p^p \chi _{\{u>0\}},\qquad 1<p<\infty , \] on open subsets $\Omega \subset \mathbb {R}^n$. There is a critical dimension $k^*$ such that the free boundary $\partial \{u>0\}\cap \Omega$ has no singularities and is a real analytic hypersurface if $p=2$ and $n<k^*$. A corollary of the main result in this note ensures that there exists $\epsilon _0>0$ such that the same result holds if $|p-2|<\epsilon _0$.References
- 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
- Luis A. Caffarelli, David Jerison, and Carlos E. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence, RI, 2004, pp. 83–97. MR 2082392, DOI 10.1090/conm/350/06339
- Donatella Danielli and Arshak Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations 23 (2005), no. 1, 97–124. MR 2133664, DOI 10.1007/s00526-004-0294-5
- Donatella Danielli and Arshak Petrosyan, Full regularity of the free boundary in a Bernoulli-type problem in two dimensions, Math. Res. Lett. 13 (2006), no. 4, 667–681. MR 2250499, DOI 10.4310/MRL.2006.v13.n4.a14
- Daniela De Silva and David Jerison, A singular energy minimizing free boundary (2005), preprint.
- David Kinderlehrer, Louis Nirenberg, and Joel Spruck, Régularité dans les problèmes elliptiques à frontière libre, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 24, A1187–A1190 (French, with English summary). MR 501110
- Sandra Martínez and Noemi Wolanski, A minimum problem with free boundary in Orlicz spaces (2006), preprint.
- James Simons, Minimal cones, Plateau’s problem, and the Bernstein conjecture, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 410–411. MR 216387, DOI 10.1073/pnas.58.2.410
- Georg Sebastian Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 (1999), no. 2, 317–326. MR 1759450, DOI 10.1007/BF02921941
Additional Information
- Arshak Petrosyan
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 654444
- Email: arshak@math.purdue.edu
- Received by editor(s): March 6, 2006
- Published electronically: March 21, 2008
- Additional Notes: The author was supported in part by NSF grant DMS-0401179.
- Communicated by: David S. Tartakoff
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2763-2769
- MSC (2000): Primary 35R35
- DOI: https://doi.org/10.1090/S0002-9939-08-09476-8
- MathSciNet review: 2399040