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Remarks on the strong maximum principle for nonlocal operators

Abstract : In this note, we study the existence of a strong maximum principle for the nonlocal operator M(x) := integral(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 is an element of 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 M. We also revisit the classical case of the convolution operator (i.e. G - (R-n, +), X = R-n, d mu = dy)
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Jerome Coville. Remarks on the strong maximum principle for nonlocal operators. Electronic Journal of Differential Equations, Texas State University, Department of Mathematics, 2008, on-line, 10 p. ⟨hal-02658335⟩

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