Remarks on the strong maximum principle for nonlocal operators

Coville, Jerome

Electronic Journal of Differential Equations, Tome 2008 (2008).

Résumé

Summary: 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. $G=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu =dy)$.

Détail

Classification : 35B50, 47G20, 35J60
Keywords: nonlocal diffusion operators, maximum principles, geometric condition