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MiS Preprint

Some remarks on the strong maximum principle arising in nonlocal operators

Pascal Autissier and Jerome Coville


In this note we make some remarks concerning maximum principles holding for nonlocal diffusion operator of the form $$\mathcal{M}[u](x) :=\int_{G}J(g)u(x*g^{-1})d\mu(g) - u(x),$$ where $G$ is a group acting continuously on a Hausdorff space $X$ and $u\in C(X)$.

We first investigate the existence of a strong maximum principle in the general situation and then focus on the case of homogeneous spaces. Depending on the topology of the homogenerous space, we give contidions on $J$ and $d\mu$ such that $\mathcal{M}$ achieves a strong maximum principle. We also revisit the classical case of convolution operator on $\mathbb{R}^n$.

Jan 25, 2008
Jan 25, 2008
MSC Codes:
47B34, 47B65, 45P05, 35B50
Nonlocal operators, Hausdorff space, Strong maximum principle

Related publications

2008 Journal Open Access
Jérôme Coville

Remarks on the strong maximum principle for nonlocal operators

In: Electronic journal of differential equations, 2008 (2008) 66, pp. 1-10