
Journal of Convex Analysis 24 (2017), No. 2, [final page numbers not yet available] Copyright Heldermann Verlag 2017 New Variational Principles of Symmetric Boundary Value Problems Abbas Moameni School of Mathematics and Statistics, Carleton University, Ottawa, Ont. K1S 5B6, Canada momeni@math.carleton.ca [Abstractpdf] The objective of this paper is to establish new variational principles for symmetric boundary value problems. Let $V$ be a Banach space and $V^*$ its topological dual. We shall consider problems of the type $\Lambda u=D \Phi(u)$ where $\Lambda: V \to V^*$ is a linear operator and $\Phi: V \to \mathbb{R}$ is a G\^ateaux differentiable convex function whose derivative is denoted by $D\Phi$. It is established that solutions of the latter equation are associated with critical points of functions of the type $$ I_{\lambda, \mu}(u):= \mu \Phi^* (\Lambda u)\lambda \Phi(u) \frac{\mu\lambda}{2}\langle \Lambda u, u \rangle, $$ where $\lambda, \mu$ are two real numbers, $\Phi^*$ is the Fenchel dual of the function $\Phi$ and $\langle .,.\rangle$ is the duality pairing between $V$ and $V^*$. By assigning different values to $\lambda$ and $\mu$ one obtains variety of new and classical variational principles associated to the equation $\Lambda u=D \Phi(u)$. Namely, EulerLagrange principle (for $\mu=0$, $\lambda=1$ and symmetric $\Lambda$), ClarkeEkeland least action principle (for $\mu=1$, $\lambda=0$ and symmetric $\Lambda$), BrezisEkeland variational principle ($\mu=1$, $\lambda=1$) and of course many new variational principles such as $$ I_{1,1}(u)= \Phi^* (\Lambda u) \Phi(u), $$ which corresponds to $\lambda=1$ and $\mu=1$. These new potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. [ Fulltextpdf (282 KB)] for subscribers only. 