Electron. J. Diff. Eqns., Vol. 2008(2008), No. 66, pp. 1-10.

Remarks on the strong maximum principle for nonlocal operators

Jérôme Coville

Abstract:
In this note, we study the existence of a strong maximum principle for the nonlocal operator
$$ \mathcal{M}[u](x) :=\int_{G}J(g)u(x*g^{-1})d\mu(g) - u(x), $$
where $G$ is a topological group acting continuously on a Hausdorff space $X$ and $u \in C(X)$. First we investigate the general situation and derive a pre-maximum principle. Then we restrict our analysis to the case of homogeneous spaces (i.e., $ X=G /H$). For such Hausdorff spaces, depending on the topology, we give a condition on $J$ such that a strong maximum principle holds for $\mathcal{M}$. We also revisit the classical case of the convolution operator (i.e. $G=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu =dy$).

Submitted January 25, 2008. Published May 1, 2008.
Math Subject Classifications: 35B50, 47G20, 35J60.
Key Words: Nonlocal diffusion operators; maximum principles; Geometric condition.

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Jérôme Coville
Max Planck Institute for mathematical science
Inselstrasse 22, D-04103 Leipzig, Germany
email: coville@mis.mpg.de

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