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On the full regularity of the free boundary in a class of variational problems

Author: Arshak Petrosyan
Journal: Proc. Amer. Math. Soc. 136 (2008), 2763-2769
MSC (2000): Primary 35R35
Published electronically: March 21, 2008
MathSciNet review: 2399040
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Abstract: We consider nonnegative minimizers of the functional

$\displaystyle J_p(u;\Omega)=\int_\Omega \vert\nabla u\vert^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 $ \vert p-2\vert<\epsilon_0$.

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Additional Information

Arshak Petrosyan
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Keywords: Regularity of the free boundary, degenerate/singular variational problem, Bernstein-type theorem, improvement of flatness
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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