Remarks on space-time variational formulations for BIEs related to the wave problem

Boundary-value problems of mathematical physics and engineering can be reformulated in terms of boundary integral equations (BIEs) knowing a fundamental solution of the involved differential operator. If in the elliptic case a wide literature, covering both theoretical properties and numerical approximation, is well established, the mathematical treatment of boundary integral equations for time-dependent problems is not so deeply investigated. Here we consider a Dirichlet (or Neumann) problem for a temporally homogeneous wave equation, reformulated as a boundary integral equation with retarded potential. So far, by means of Laplace transform in time variable, the mathematical study of this BIE has followed the lines of elliptic theory (Helmoltz equation), and in this direction the most interesting results are given by the weak formulation due to Bamberger and Ha Duong. Anyway, we believe that the following two issues are of interest: i) the exploitation of a functional approach where space and time remain coupled, owing to the optimal well-posedness results of the considered problem, which will lead to a suitable weak formulation; ii) the introduction of (extended) variational formulations, which allow the definition of suitable functionals whose minimum (or saddle) point is solution of the given BIE. We will introduce and analyze the above issues, from both theoretical and numerical point of view, referring to a one-dimensional model problem.