Uber das dirichletsche rand- und anfangswertproblem fur hyper-boiische differentialgleichungen†

Abstract

We study the- initial and boundary value problem for the equation u_tt = L[u] with Dirichlet boundary data where L is the sum of a strongly elliptic formally self-adjoint differential operator of order 2m and of an arbitrary differential operator of order q ≦m. We give a weak formulation of this problem in an appropriate Hilbert space. The main ideas of the existence, uniqueness and regularity theory are sketched. Detailed proofs will appear in two subsequent papers. Our existence argument makes essential use of L₂-spaces of vector-valued functions with values in a Hilbert space. The theory of these "Bochner spaces" can be developed by employing methods of L. Schwartz's theory of distributions. In particular, our analysis avoids Lebesgue and Bochner integration