A NOTE ON GENERALIZED TRAVELING WAVE SOLUTIONS FOR MEAN CURVATURE FLOWS WITH MERELY L ∞ FORCING IN THE PLANE
Abstract
In this note, we announce some results whose proofs are the object of a forthcoming article. We consider traveling wave solutions of curvature flows with periodic forcing R. The fixed front's profile is the supergraph of a function ψ. The latter satisfies a PDE whose existence of bounded solutions typically holds when min R > 0. This theory extends to generalized traveling wave solutions for R ∈ W 1,∞ with positive mean value. The front's profile is then unbounded and the new formulation is variational. Equivalently ψ solves a boundary value problem with infinite Dirichlet condition, on a certain domain to be determined. Our analysis concerns merely L ∞ forcing. In that case, the front's profile can have vertical lines. This amounts to consider discontinuous ψ with interface-like conditions at singularities. We then compute all possible front's profiles for a prototypical fibered medium, identifying e.g. necessary and sufficient conditions for their boundedness. We also shed light on the possible failure of a certain uniqueness property, which was typical of the W 1,∞ theory.
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