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# Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary

• 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.
Mathematics Subject Classification: Primary: 49L25; Secondary: 53C44.

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