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Zeitschrift für Analysis und ihre Anwendungen

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Volume 29, Issue 3, 2010, pp. 309–350
DOI: 10.4171/ZAA/1411

Published online: 2010-06-23

Non-Local Hamilton-Jacobi Equations Arising in Dislocation Dynamics

Hitoshi Ishii[1] and Yutaka Matsumura[2]

(1) Waseda University, Tokyo, Japan
(2) Waseda University, Tokyo, Japan

We investigate a class of non-local Hamilton–Jacobi equations arising in dislocation dynamics. The class of Hamilton–Jacobi equations treated here is a variation of those studied by N. Forcadel, C. Imbert and R. Monneau in [Discrete Contin. Dyn. Syst. 23 (2009)(3), 785 – 826], and the new feature lies in the singularity at the origin of the kernel functions which describe non-local effects. For the class of Hamilton–Jacobi equations, we establish some stability properties of (viscosity) solutions, comparison theorems between subsolutions and supersolutions and existence theorems of solutions.

Keywords: Hamilton–Jacobi equations, functional differential equations, dislocation dynamics, viscosity solutions

Ishii Hitoshi, Matsumura Yutaka: Non-Local Hamilton-Jacobi Equations Arising in Dislocation Dynamics. Z. Anal. Anwend. 29 (2010), 309-350. doi: 10.4171/ZAA/1411