Variational Formulations

In this chapter we will discuss the variational formulation for boundary integral equations and its connection to the variational solution of partial differential equations. We collect some basic theorems in functional analysis which are needed for this purpose. In particular, Green's theorems and the Lax–Milgram theorem are fundamental tools for the solvability of boundary integral equations as well as for elliptic partial differential equations. We will present here a subclass of boundary value problems for which the coerciveness property for some associated boundary integral operators follows directly from that of the variational form of the boundary and transmission problems. In this class, the solution of the boundary integral equations will be established with the help of the existence and regularity results of elliptic partial differential equations in variational form. This part of our presentation goes back to J.C. Nedelec and J. Planchard [235] and is an extension of the approach used by J.C. Nedelec [231] and the ”French School” (see Dautray and Lions [59, Vol. 4]). It also follows closely the work by M. Costabel [49, 50, 51] and our works [55, 138, 292].