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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

A problem with an obstacle that goes out to the boundary of the domain for a class of quadratic functionals on $ \mathbb{R}^N$


Author: A. A. Arkhipova
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 847-875
MSC (2010): Primary 35J20
Published electronically: August 18, 2011
MathSciNet review: 2798764
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Abstract: A variational problem with obstacle is studied for a quadratic functional defined on vector-valued functions $ u : \Omega\to \mathbb{R}^N, N>1$. It is assumed that the nondiagonal matrix that determines the quadratic form of the integrand depends on the solution and is ``split''. The role of the obstacle is played by a closed (possibly, noncompact) set $ \mathcal{K}$ in $ \mathbb{R}^N$ or a smooth hypersurface $ S$. It is assumed that $ u(x)\in\mathcal{K}$ or $ u(x)\in S$ a.e. on $ \Omega$. This is a generalization of a scalar problem with an obstacle that goes out to the boundary of the domain. It is proved that the solutions of the variational problems in question are partially smooth in $ \widebar{\Omega}$ and that the singular set $ \Sigma$ of the solution satisfies $ H_{n-2}(\Sigma)=0$.


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Additional Information

A. A. Arkhipova
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
Email: arina@AA1101.spb.edu

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01172-0
PII: S 1061-0022(2011)01172-0
Keywords: Variational problem, quadratic functional, nondiagonal matrix, Signorini condition
Received by editor(s): April 7, 2010
Published electronically: August 18, 2011
Additional Notes: Supported by RFBR (grant no. 09-01-00729) and by the grant NSH-4210.2010.1 for support of leading scientific schools
Dedicated: Dedicated to Vasiliĭ Mikhaĭlovich Babich
Article copyright: © Copyright 2011 American Mathematical Society