Generalized junction conditions for degenerate parabolic equations
Cyril Imbert, Vinh Duc Nguyen · arXiv (Cornell University) · 2016
In this paper, we study degenerate parabolic equations in non-divergence form whose coefficients are discontinuous along interfaces. At these discontinuities, the imposed conditions are compatible with the maximum principle but they do not rely on any compatibility of coefficients across interfaces. An important and simple example is when the equation is posed in the Euclidian space and when coefficients are smooth on either side of a hyperplane. There are also some motivations for solving such partial differential equations on networks, and in particular on junctions, that are the networks made of one vertex and a finite number of infinite edges. We explain here that the approach proposed by the first author 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, which are of control-type. We then use the vertex test function (Imbert, Monneau – 2014) to prove a comparison principle. These results extend naturally to the multi-dimensional setting and to stationary solutions. We apply our results to two different multi-dimensional problems. On the one hand, we give a complete answer to an open question about these equations: we 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. On the other hand, we give a short and simple PDE proof of a large deviation results of Boue, Dupuis and Ellis (2000).