One dimensional infinite-horizon variational problems arising in continuum mechanics

Notes on a class of one-dimensional Landau-Brazovsky models

“…According to Theorem [1], it is known that problem (P # ) has a minimizer ω. Combining with symmetric properties of ω, it remains to show that ω has a zero.…”

“…Worth mentioning, Zaslavski [6] generalized [1]'s result and proved the existence of a nonconstant periodic solution under the condition that µ < inf{f (t, 0, 0)|t ∈ R} for a class of functions f . In our situation, either [1] or [6] leads to the same conclusion. The proofs are based on the methods developed by V. J. Mizel, L. A. Peletier, and W. C. Troy [5,3].…”

On the thermodynamics of periodic phases

“…Thus, every point at which Hf is not differentiable is an exposed point for Concerning exposed points for *P, we have the following lemma, which holds for general materials obeying the hypotheses of Section 2, once it is granted that <!>, and hence «P, is real-valued. 1) -(3.4), the Lemma just proven and the results of Leizarowitz and Mizel [1989] inply that when a is an exposed point for VP there is a periodic state u a that solves the problem (P fl ).…”

“…is continuous on J g . Now, in view of the theory presented in [1989] and the previously mentioned result of n (n) Leizarowitz [1990], we can choose the functions n (n) that solve (P*") so that they are Proofs and applications of the above observations will be presented elsewhere.…”

“…For our next lemma we employ the following result, recently obtained by Leizarowitz [1990] using a refinement of arguments given in [1989]: Let I be a bounded interval. As X varies over J, for each X a periodic function u x that solves (P*) can be selected so that (i) the set { T k j X € / }, with T x the minimal period ofu x , is bounded, and (ii) the functions u x and u x% Xel t are uniformly bounded.…”