On fractional higher-order Dirichlet boundary value problems: between the Laplacian and the bilaplacian

TL;DR

This paper explores fractional higher-order Dirichlet boundary value problems, bridging the gap between Laplacian and bilaplacian operators, by examining solution behaviors, boundary conditions, and maximum principles through variational and explicit kernel methods.

Contribution

It introduces a framework for understanding intermediate fractional Laplacian problems, including boundary conditions, solution properties, and limiting behaviors, expanding the theoretical understanding of higher-order PDEs.

Findings

Solutions are obtained variationally and explicitly for the ball.

Maximum principles are analyzed for intermediate fractional problems.

Behavior of solutions approaches Laplacian or bilaplacian cases in limits.

Abstract

The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order problems are less rigid and more complex. One way to better understand this transition is to study the intermediate Dirichlet problem in terms of fractional Laplacians. This survey aims to be an introduction to these type of problems; in particular, the different pointwise notions for these operators is introduced considering a suitable natural extension of the Dirichlet boundary conditions for the fractional setting. Solutions are obtained variationally and, in the case of the ball, via explicit kernels. The validity of maximum principles for these intermediate problems is also discussed as well as the limiting behavior of solutions when approaching the Laplacian or the bilaplacian case.