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Uniform estimates for a variational problem with small parameters. (English) Zbl 0793.49019

The author studies the constrained variational problem, \(\inf\{J_ I[u]: u\in H_ 2(I), \langle u\rangle_ I= a\}\), where \(I\) is a bounded interval on the line and \[ J_ I[u]= {1\over | I|} \int_ I (u''{}^ 2- \mu u'{}^ 2+ \psi(u))dt\quad\text{and}\quad \langle u\rangle_ I= {1\over | I|} \int_ I u dt. \] Here \(\mu\) is a positive number and \(\psi\) is a double well potential, e.g. \(\psi(u)=(u^ 2- 1)^ 2\). This problem (which we denote by \((P^ a_ I)\)) was introduced in B. D. Coleman, M. Marcus and V. J. Mizel [Arch. Ration. Mech. Anal. 117, No. 4, 321-347 (1992)] as a model for the determination of the thermodynamical equilibrium states of unidimensional bodies. In the above-mentioned paper the authors were interested in studying the patterns of equilibrium states of large bodies. For this purpose they investigated a version of the above model in which the underlying domain is the whole line. In this version one defines \(J_ R[u]\) (the energy of a state \(u\in H^{\text{loc}}_ 2(R)\)) as \(\liminf_{T\to\infty} J_{(-T,T)}[u]\) and the average mass \(\langle u\rangle_ R\) as the limit of \(\langle u\rangle_{(-T,T)}\) (which is assumed to exist). In the present paper the author investigates the relation between the formally limiting problem \((P^ a_ R)\) and the problems \((P^ a_ I)\) as \(| I|\to\infty\). The main part of the paper is devoted to the derivation of uniform a priori estimates for equilibrium states of problem \((P^ a_ I)\), which are crucial to this investigation.
Reviewer: M.Marcus (Haifa)

MSC:

49S05 Variational principles of physics
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References:

[1] Adams, R. A., Sobolev Spaces, Academic Press, 1975.
[2] Attouch, H., Variational Convergence for Functions and Operators, Pitman, 1984. · Zbl 0561.49012
[3] Coleman, B. D., Marcus, M. & Mizel, V. J., On the thermodynamics of periodic phases, Arch. Rational Mech. Anal. 117 (1992), 321-347. · Zbl 0788.73015
[4] Leizarowitz, A., Infinite horizon autonomous systems with unbounded cost, Appl. Math. Optim. 13 (1985), 19-43. · Zbl 0591.93039
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[6] Marcus, M. & Mizel, V. J., Higher order variational problems related to a model in thermodynamics (in preparation).
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