A variational principle associated with a certain class of boundary-value problems

Abstract

A variational principle is introduced to provide a new formulation and resolution for several boundary-value problems. Indeed, we consider systems of the form \begin{eqnarray*} \left\{ \begin{array}{ll} \Lambda u = \nabla \Phi (u), \\ \beta_2 u= \nabla \Psi (\beta_1 u), \end{array} \right. \end{eqnarray*} where $\Phi$ and $\Psi$ are two convex functions and $\Lambda$ is a possibly unbounded self-adjoint operator modulo the boundary operator ${\mathcal B}= (\beta_1, \beta_2).$ We shall show that solutions of the above system coincide with critical points of the functional $$ I(u)= \Phi^* (\Lambda u)-\Phi(u)+\Psi^* (\beta_2 u)- \Psi(\beta_1 u), $$ where $\Phi^*$ and $\Psi^*$ are the Fenchel-Legendre dual of $\Phi$ and $\Psi$ respectively. Note that the standard Euler-Lagrange functional corresponding to the system above is of the form, $$ F(u)= \tfrac{1}{2} \langle \Lambda u, u \rangle- \Phi(u)-\Psi (\beta_1 u). $$ An immediate advantage of using the functional $I$ instead of $F$ is to obtain more regular solutions and also the flexibility to handle boundary-value problems with nonlinear boundary conditions. Applications to Hamiltonian systems and semi-linear Elliptic equations with various linear and nonlinear boundary conditions are also provided.