Some results on stochastic homogenization of non-convex Hamilton-Jacobi equations

William Feldman (Chicago, USA)

I will discuss some progress about the homogenization of non-convex Hamilton-Jacobi equations in random media.  I will revisit the recent counter-example of Ziliotto who constructed a coercive but non-convex Hamilton-Jacobi equation with stationary ergodic random potential field for which homogenization does not hold.  We have extended this result showing that for any Hamiltonian with a strict saddle-point there is a random stationary ergodic potential field V so that homogenization does not hold for the Hamiltonian H = h(p)-V(x).  I will also discuss a positive result,  under a finite range of dependence assumption we show that homogenization holds for Hamiltonians with strictly star-shaped sub-level sets.

This talk is based on joint work with P. Souganidis.

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