Boggio's formula for fractional polyharmonic Dirichlet problems

TL;DR

This paper extends Boggio's formula to fractional polyharmonic Dirichlet problems, providing a unified approach that applies to a broader range of fractional parameters in ball domains.

Contribution

It introduces a consistent formulation of Boggio's formula for fractional polyharmonic problems, generalizing existing solutions for fractional parameters less than 1.

Findings

Boggio's formula is extended to fractional polyharmonic problems.

The formulation applies to all fractional parameters, not just less than 1.

Solutions are explicitly constructed for fractional Dirichlet problems in balls.

Abstract

Boggio's formula in balls is known for integer-polyharmonic Dirichlet problems and for fractional Dirichlet problems with fractional parameter less than 1. We give here a consistent formulation for fractional polyharmonic Dirichlet problems such that Boggio's formula in balls yields solutions also for the general fractional case.