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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2016
- DOI:
- 10.48550/arXiv.1607.00929
- arXiv:
- arXiv:1607.00929
- Bibcode:
- 2016arXiv160700929A
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- Revised version