2017/04/13 by Pierre‐Louis Lions, Panagiotis E. Souganidis, Lions, Pierre-Louis +1 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1704.04001
We consider multi-dimensional junction problems for first- and second-order\npde with Kirchoff-type Neumann boundary conditions and we show that their\ngeneralized viscosity solutions are unique. It follows that any viscosity-type\napproximation of the junction problem converges to a unique limit. The results\nhere are the first of this kind and extend previous work by the authors for\none-dimensional junctions. The proofs are based on a careful analysis of the\nbehavior of the viscosity solutions near the junction, including a blow-up\nargument that reduces the general problem to a one-dimensional one. As in our\nprevious note, no convexity assumptions and control theoretic interpretation of\nthe solutions are needed.\n