Direct Methods in Equilibrium Theory

Abstract

This chapter treats the equilibrium states by the direct methods of the calculus of variations. Given a class of admissible rest states Σ ₀ satisfying various constraints (such as the boundary conditions), one seeks a state σ ₀ for which the total canonical free energy P takes the least possible value on Σ ₀. This state need not exist, depending on the free energy $bar y$ and on the class Σ ₀. However, if the total energy is bounded from below on Σ ₀, then P₀ := inf {P(σ): σ ∈ Σ ₀} is finite and one can always find a sequence σ _k ∈ Σ ₀ such that P(σ _k ) → P₀ as k → ∞, the minimizing sequence. Under the condition of coercivity, one can find a subsequence, still denoted by σ _k , which converges weakly to some state σ ₀ and we assume that σ ₀ ∈ Σ ₀. (To guarantee this inclusion, one has to admit states with a lower degree of smoothness than, say, continuous differentiability, see Sect. 21.2.) Since σ _k converges weakly to σ ₀ and P(σ ₀) approaches P₀, a natural question is whether P₀ = P(σ ₀). This will be the case if P is sequentially weakly lower semicontinuous (swlsc), i.e., if for every sequence σ _k in Σ ₀ converging weakly to some σ ∈ Σ ₀, we have

$$\mathop {\lim \inf {\rm P}}\limits_{k \to \infty } \left( {{\sigma _k}} \right) \geqslant {\rm P}{\left( \sigma \right)_ \cdot }$$

.