JournalsrlmVol. 28, No. 4pp. 807–816

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

  • Pierre-Louis Lions

    Collège de France, Paris, and Université de Paris-Dauphine, France
  • Panagiotis E. Souganidis

    University of Chicago, USA
Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions cover

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Abstract

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.

Cite this article

Pierre-Louis Lions, Panagiotis E. Souganidis, Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 4, pp. 807–816

DOI 10.4171/RLM/786