Complementary variational principles for ∇2φ = f(φ) with applications to the Thomas–Fermi and Liouville equations

Abstract

Complementary variational principles associated with the boundary-value problem ∇²φ = f(φ) in ∇,φ = φ₀ on the boundary of ∇, are presented. The theory is applied to the Thomas–Fermi equation in quantum mechanics and the Liouville equation in equilibrium statistical mechanics.