Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 66, pp. 1-10.
Title: Remarks on the strong maximum principle for
nonlocal operators
Author: Jerome Coville (Max Planck Institute, Leipzig, Germany)
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 01, 2008.
Math Subject Classifications: 35B50, 47G20, 35J60
Key Words: Nonlocal diffusion operators; maximum principles;
Geometric condition.