Well posedness for multi-dimensional junction problems with Kirchoff-type conditions

By Pierre-Louis Lions, Panagiotis E. Souganidis. 2017

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We consider multi-dimensional junction problems for first- and second-order pde with Kirchoff-type Neumann boundary conditions and we show that their generalized viscosity solutions are unique. It follows that any viscosity-type approximation of the junction problem converges to a unique limit. The results here are the first of this kind and extend previous work by the authors for one-dimensional junctions. The proofs are based on a careful analysis of the behavior of the viscosity solutions near the junction, including a blow-up argument that reduces the general problem to a one-dimensional one. As in our previous note, no convexity assumptions and control theoretic interpretation of the solutions are needed.

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/41f143818f2f6878e83a8195d97faf405a5e97b8