- journal article metadata
European Mathematical Society Publishing House
2017-11-18 23:40:03
Rendiconti Lincei - Matematica e Applicazioni
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
RLM
1120-6330
1720-0768
General
10.4171/RLM
http://www.ems-ph.org/doi/10.4171/RLM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
28
2017
4
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
Souganidis
University of Chicago, USA
Hamilton–Jacobi equations, networks, discontinuous Hamiltonians, junction problesm, stratification problems, comparison principle, viscosity solutions
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.
Partial differential equations
Calculus of variations and optimal control; optimization
807
816
10.4171/RLM/786
http://www.ems-ph.org/doi/10.4171/RLM/786
11
15
2017