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

Abstract

In the paper, a class of one-dimensional Landau-Brazovsky models is investigated. We present a sufficient condition under which the corresponding functional achieves its minimum. Moreover, a nonexistence result for nontrivial critical points is given. (2000). Primary 34B15; Secondary 49J99. Mathematics Subject Classification

Nontrivial Periodic Minimizer for Landau-Brazovskii Model with Constraint

“…Furthermore, since ̟n solves the E-L equation ̟′′′′ n + ̟′′ n + ̟n + h ′ ( ̟n ) = 0. By (8) and the boundedness of λ n , ̟n must be bounded in W Proof. Since the functional I T 1 ,T 2 (f, ̟) is independent of the time variable t, we may assume ̟n (0) = min s∈R ̟n (s) , ∀n.…”

“…In [8], the model ( 1) with symmetric double well potential is studied and the existence of global periodic minimizer is shown. The author also proved the symmetric property of the minimizer.…”

“…In addition, without the presence of symmetric property, a non-existence result is given there. In this paper, we study the constrained version of [8]'s model, but we do not assume any symmetric properties of (1). The proof is given following that of [6], however, we are interested in the existence of nontrivial periodic minimizer, a sufficient condition for the existence is given.…”

Heteroclinic Solutions for Nonautonomous EFK Equations

“…In spite of its practical significance [14], there has been few researches on (1) with time-dependent potentials; the paper seems to be the first attempt in the direction and the methods here can be applied to similar equations.…”