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Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case

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  • A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic penalization function |x-y|2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate.

    Mathematics Subject Classification: 35F21, 49L25, 35B51.


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  • Figure 1.  A Hamilton-Jacobi equation posed on a multi-dimensional junction. Here there are 3 branches (or sheets — $N=3$ ) and the tangential dimension is $1$ ( $d=1$ ). We did not illustrate the junction condition on the junction hyperplane $\Gamma$ (which is a line in this example)

    Figure 2.  Monotone parts $H_i^\pm$ of a Hamiltonian $H_i$ ($H_i^-$ on the left, $H_i^+$ on the right). The Hamiltonian is in black, monotone parts in red. The tangent variable $p'$ is not shown. In this example, the minimum $A_i$ of $H_i$ is lower than $A_0$. The "inverse" functions $\pi_i^\pm$ of $H_i$ are also shown

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