On the existence of the biharmonic Green kernels and the the adjoint biharmonic functions

TL;DR

This paper investigates the existence and regularity of biharmonic Green kernels in specific spaces, providing examples of non-existence, and introduces the adjoint biharmonic space to clarify previous incomplete results.

Contribution

It offers new insights into the existence conditions of biharmonic Green kernels and develops the theory of the adjoint biharmonic space, clarifying prior incomplete results.

Findings

Biharmonic Green kernels do not always exist in certain spaces.

Introduction of the adjoint biharmonic space clarifies previous incomplete results.

Provides examples illustrating the non-existence of the Green kernel.

Abstract

We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.