## 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:

  Adams, R. A., Sobolev Spaces, Academic Press, 1975.  Attouch, H., Variational Convergence for Functions and Operators, Pitman, 1984. · Zbl 0561.49012  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  Leizarowitz, A., Infinite horizon autonomous systems with unbounded cost, Appl. Math. Optim. 13 (1985), 19-43. · Zbl 0591.93039  Leizarowitz, A. & Mizel, V. J., One dimensional infinite-horizon variational problems arising in continuum mechanics, Arch. Rational. Mech. Anal. 106 (1989), 161-194. · Zbl 0672.73010  Marcus, M. & Mizel, V. J., Higher order variational problems related to a model in thermodynamics (in preparation).
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