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Effective junction conditions for degenerate parabolic equations

Abstract : We are interested in the study of parabolic equations on a multi-dimensional junction, i.e. the union of a finite number of copies of a half-hyperplane of dimension d + 1 whose boundaries are identified. The common boundary is referred to as the junction hyperplane. The parabolic equations on the half-hyperplanes are in non-divergence form, fully non-linear and possibly degenerate, and they do degenerate and are quasi-convex along the junction hyperplane. More precisely, along the junction hyperplane the non-linearities do not depend on second order derivatives and their sublevel sets with respect to the gradient variable are convex. The parabolic equations are supplemented with a non-linear boundary condition of Neumann type, referred to as a generalized junction condition, which is compatible with the maximum principle. Our main result asserts that imposing a generalized junction condition in a weak sense reduces to imposing an effective one in a strong sense. This result extends the one obtained by Imbert and Monneau for Hamilton-Jacobi equations on networks and multi-dimensional junctions. We give two applications of this result. On the one hand, we give the first complete answer to an open question about these equations: we prove in the two-domain case that the vanishing viscosity limit associated with quasi-convex Hamilton-Jacobi equations coincides with the maximal Ishii solution identified by Barles, Briani and Chasseigne (2012). On the other hand, we give a short and simple PDE proof of a large deviation result of Boué, Dupuis and Ellis (2000).
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Submitted on : Monday, September 18, 2017 - 5:04:27 PM
Last modification on : Thursday, April 23, 2020 - 10:18:02 AM

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Cyril Imbert, Vinh Duc Nguyen. Effective junction conditions for degenerate parabolic equations. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2017, 56 (157), ⟨10.1007/s00526-017-1239-0⟩. ⟨hal-01252891v3⟩

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