Positive powers of the Laplacian: from hypersingular integrals to boundary value problems

TL;DR

This paper explores positive powers of the Laplacian through hypersingular integrals, offering a flexible pointwise evaluation method applicable to boundary value problems and connecting these to polyharmonic operators via asymptotic analysis.

Contribution

It introduces a new pointwise evaluation for positive Laplacian powers, extending applicability to boundary value problems and establishing a variational framework.

Findings

Pointwise evaluation applicable to general boundary problems

Explicit examples demonstrating the method

Reduction to polyharmonic operators in certain cases

Abstract

Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-contained as possible and relies on elementary pointwise calculations and known identities for special functions.