On the maximum principle for higher-order fractional Laplacians

TL;DR

This paper investigates the properties of solutions to higher-order fractional Laplacian equations, revealing limitations of maximum principles and establishing regularity and positivity results using advanced potential theory and representation formulas.

Contribution

It provides explicit counterexamples to maximum principles for certain fractional orders and extends regularity and positivity results to higher-order fractional Laplacians.

Findings

Counterexample to maximum principles for $s otin ext{integers}$

Regularity and positivity preservation in whole space and ball

Validation of Boggio's representation formula for all $s>0$

Abstract

We study existence, regularity, and qualitative properties of solutions to linear problems involving higher-order fractional Laplacians $(-\Delta)^s$ for any $s>1$. Using the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for $s\in(n,n+1)$ with $n\in\mathbb N$ odd; moreover, using a representation formula for solutions, we derive regularity and positivity preserving properties whenever the domain is the whole space or a ball. In the case of the whole space we analyze the Riesz kernel, which provides a fundamental solution, while in the case of the ball we show the validity of Boggio's representation formula for all integer and fractional powers of the Laplacian $s>0$. Our proofs rely on characterizations of $s$-harmonic functions using higher-order Martin kernels, on a decomposition of Boggio's formula, and on elliptic regularity theory.