Schwartz-type integrals in a biharmonic plane

TL;DR

This paper develops explicit Schwartz-type integral solutions for boundary value problems related to the biharmonic equation within a specialized algebraic framework, demonstrating solvability conditions for different domains.

Contribution

It introduces explicit Schwartz-type integral solutions for biharmonic boundary value problems in a biharmonic algebra setting, establishing solvability criteria for half-plane and disk domains.

Findings

Solutions are explicit via Schwartz-type integrals.

Unconditional solvability in a half-plane.

Solvability in a disk depends on a natural condition.

Abstract

We consider a two-dimensional commutative algebra B over the field of complex numbers. The algebra B is associated with the biharmonic equation. For monogenic functions with values in B, we consider a Schwartz-type boundary value problem (associated with the main biharmonic problem) for a half-plane and for a disk of the biharmonic plane. We obtain solutions in explicit forms by means of Schwartz-type integrals and prove that the mentioned problem is solvable unconditionally for a half-plane but it is solvable for a disk if and only if a certain natural condition is satisfied.