On the maximum principle for higher-order fractional Laplacians

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.