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Stochastic homogenization of interfaces moving with changing sign velocity. (English) Zbl 1308.35024
Summary: We are interested in the averaging behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign. The problem can be reformulated as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous of degree one non-coercive Hamiltonian. The periodic setting was studied earlier by P. Cardaliaguet et al. [J. Math. Pures Appl. (9) 91, No. 4, 339–363 (2009; Zbl 1180.35070)]. Here, we concentrate in the random media and show that the solutions of the oscillatory Hamilton-Jacobi equation converge in \(L^\infty\)-weak \(\star\) to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
70H20 Hamilton-Jacobi equations in mechanics
37A50 Dynamical systems and their relations with probability theory and stochastic processes
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
78A48 Composite media; random media in optics and electromagnetic theory
35R60 PDEs with randomness, stochastic partial differential equations
35F21 Hamilton-Jacobi equations
35D40 Viscosity solutions to PDEs
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