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Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level
- Publication Year :
- 2018
-
Abstract
- We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $\Omega$ of $\R^n$ assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in $\Omega$. Then, we analyse the singular set of a solution showing that singularities propagate along suitable curves, the so-called generalized characteristics, and that such curves stay singular unless they reach the boundary of $\Omega$. Moreover, we prove that the latter is never the case for mechanical systems and that singular generalized characteristics converge to a critical point of the solution in finite or infinite time. Finally, under stronger assumptions for the domain and Dirichlet data, we are able to conclude that solutions are globally semiconcave and semiconvex near the boundary.
- Subjects :
- Mathematics - Analysis of PDEs
Mathematics - Dynamical Systems
35F21, 49L25, 37J50
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1803.01591
- Document Type :
- Working Paper