Laplace transformation of vector-valued distributions and applications to Cauchy-Dirichlet problems

TL;DR

This paper introduces new proofs for the Laplace transformation of vector-valued distributions and applies these results to explicitly solve Cauchy-Dirichlet problems for polyharmonic, wave, and Klein-Gordon operators in half-spaces.

Contribution

It provides novel proofs of the exchange theorem and explicit solutions to complex boundary value problems in half-spaces.

Findings

New proofs of the exchange theorem for vector-valued distributions

Explicit solutions to polyharmonic Dirichlet problems in half-spaces

Explicit solutions to wave and Klein-Gordon Cauchy-Dirichlet problems

Abstract

We present two new proofs of the exchange theorem for the Laplace transformation of vector-valued distributions. We then derive an explicit solution to the Dirichlet problem of the polyharmonic operator in a half-space. Finally, we obtain explicit solutions to Cauchy-Dirichlet problems of iterated wave- and Klein-Gordon-operators in half-spaces.