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April  2010, 26(2): 493-519. doi: 10.3934/dcds.2010.26.493

Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary

1. 

Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914

2. 

Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

3. 

Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 07-097 Warsaw

Received  January 2009 Revised  July 2009 Published  October 2009

We consider the weighted mean curvature flow in the plane with a driving term. For certain anisotropy functions this evolution problem degenerates to a first order Hamilton-Jacobi equation with a free boundary. The resulting problem may be written as a Hamilton-Jacobi equation with a spatially non-local and discontinuous Hamiltonian. We prove existence and uniqueness of solutions. On the way we show a comparison principle and a stability theorem for viscosity solutions.
Citation: Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493
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