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.

Revised version

References in corpus (5)

• Nonlocal diffusion and applications
• Some observations on the Green function for the ball in the fractional Laplace framework
• Spectral results for mixed problems and fractional elliptic operators
• Integral representation of solutions to higher-order fractional Dirichlet problems on balls
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Cited by in corpus (7)

• Integral representation of solutions to higher-order fractional Dirichlet problems on balls
• Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas
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• Starshape of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings
• Neumann conditions for the higher order $s$-fractional Laplacian $(-\Delta {)}^{s}u$ with $s>1$
• On a fractional boundary version of Talenti's inequality in the unit ball