Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

TL;DR

This paper derives explicit solutions for boundary value problems involving positive powers of the Laplacian in the half-space, including nonlocal cases, using a point inversion transformation and providing regularity and growth estimates.

Contribution

It introduces explicit formulas for non-integer powers of the Laplacian in the half-space with Dirichlet conditions, extending classical results to nonlocal operators.

Findings

Explicit formulas for solutions to boundary value problems

Extension of Dirichlet conditions for nonlocal operators

Regularity and growth estimates for solutions

Abstract

We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.