Maximum and antimaximum principles for some nonlocal diffusion operators
In this work we consider the maximum and antimaximum principles for the nonlocal Dirichlet problem J * u - u + λ u + h = ∫RN J (x - y) u (y) d y - u (x) + λ u (x) + h (x) = 0 in a bounded domain Ω, with u (x) = 0 in RN {set minus} Ω. The kernel J in the convolution is assumed to be a continuous, com...
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0362546X_v71_n12_p6116_GarciaMelian |
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| Sumario: | In this work we consider the maximum and antimaximum principles for the nonlocal Dirichlet problem J * u - u + λ u + h = ∫RN J (x - y) u (y) d y - u (x) + λ u (x) + h (x) = 0 in a bounded domain Ω, with u (x) = 0 in RN {set minus} Ω. The kernel J in the convolution is assumed to be a continuous, compactly supported nonnegative function with unit integral. We prove that for λ < λ1 (Ω), the solution verifies u > 0 in over(Ω, -) if h ∈ L2 (Ω), h ≥ 0, while for λ > λ1 (Ω), and λ close to λ1 (Ω), the solution verifies u < 0 in over(Ω, -), provided ∫Ω h (x) φ{symbol} (x) d x > 0, h ∈ L∞ (Ω). This last assumption is also shown to be optimal. The "Neumann" version of the problem is also analyzed. © 2009 Elsevier Ltd. All rights reserved. |
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