New Variational Principles of Symmetric Boundary Value Problems

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