Abstract

The objective of this paper is to establish new variational principles for symmetric boundary value problems. Let VV be a Banach space and V∗V^* its topological dual. We shall consider problems of the type Λu=DΦ(u)\Lambda u=D \Phi(u) where Λ:V→V∗\Lambda: V \to V^* is a linear operator and Φ:V→R\Phi: V \to \mathbb{R} is a Gâteaux differentiable convex function whose derivative is denoted by DΦD\Phi. It is established that solutions of the latter equation are associated with critical points of functions of the type Iλ,μ(u):=μΦ∗(Λu)−λΦ(u)−μ−λ2⟨Λu,u⟩,I_{\lambda, \mu}(u):= \mu \Phi^* (\Lambda u)-\lambda \Phi(u)- \frac{\mu-\lambda}{2}\langle \Lambda u, u \rangle, where λ,μ\lambda, \mu are two real numbers, Φ∗\Phi^* is the Fenchel dual of the function Φ\Phi and ⟨.,.⟩\langle.,.\rangle is the duality pairing between VV and V∗V^*. By assigning different values to λ\lambda and μ\mu one obtains variety of new and classical variational principles associated to the equation Λu=DΦ(u)\Lambda u=D \Phi(u). Namely, Euler-Lagrange principle (for μ=0\mu=0, λ=1\lambda=1 and symmetric Λ\Lambda), Clarke-Ekeland least action principle (for μ=1\mu=1, λ=0\lambda=0 and symmetric Λ\Lambda), Brezis-Ekeland variational principle (μ=1\mu=1, λ=−1\lambda=-1) and of course many new variational principles such as I1,1(u)=Φ∗(Λu)−Φ(u),I_{1,1}(u)= \Phi^* (\Lambda u)- \Phi(u), which corresponds to λ=1\lambda=1 and μ=1\mu=1. These new potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems

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Abbas Moameni

School of Mathematics and Statistics, Carleton University, Ottawa, Ont. K1S 5B6, Canada

momeni@math.carleton.ca

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Suggested citation

A. Moameni. “New Variational Principles of Symmetric Boundary Value Problems.” Journal of Convex Analysis 24 (2017), No. 2, 365–381. https://doi.org/10.68381/jca24025

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