Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the maximum principle for higher-order fractional Laplacians   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.

25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph