Generalized junction conditions for degenerate parabolic equations

Vinh Nguyen (Cardiff, UK)

With C. IMBERT, we study degenerate parabolic equations in multi-domains whose coefficients are discontinuous along interfaces.  We observe that the approach proposed by IMBERT and Monneau (2014) for Hamilton-Jacobi equations can be further developed to handle generalized junction conditions (such as the generalized Kirchoff ones) and second order terms. We first prove that generalized junction conditions reduce to flux-limited ones.  We then use then vertex test function (Imbert, Monneau — 2014) to prove a comparison principle.  We  then determine the vanishing viscosity limit associated with Hamilton-Jacobi equations posed on multi-domains and networks.  In the two-domain and convex case, the maximal Ishii solution identified by Barles, Briani and Chasseigne (2012) is selected. Finally, we give a short and simple PDE proof for the large deviation result of Boue, Dupuis and Ellis (2000).

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