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Journal of Convex Analysis 24 (2017), No. 2, 365--381
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


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, Euler-Lagrange principle (for $\mu=0$, $\lambda=1$ and symmetric $\Lambda$), Clarke-Ekeland least action principle (for $\mu=1$, $\lambda=0$ and symmetric $\Lambda$), Brezis-Ekeland 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.

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