A VARIATIONAL PRINCIPLE ASSOCIATED WITH A CERTAIN CLASS OF BOUNDARY-VALUE PROBLEMS

A variational principle is introduced to provide a new formulation and resolution for several boundary-value problems. Indeed, we consider systems of the form {Lambda u = del Phi(u), beta(2)u = del Psi(beta(1)u), where Phi and Psi are two convex functions and Lambda is a possibly unbounded self-adjoint operator modulo the boundary operator beta = (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(mu) = 1/2 <Lambda mu, mu > - 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-vale problems with nonlinear boundary conditions. Applications to Hamiltonian systems and semi-linear Elliptic equations with various linear and nonlinear boundary conditions are also provided.