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Global Generalized Characteristics for the Dirichlet Problem for Hamilton--Jacobi Equations at a Supercritical Energy Level
We study the nonhomogeneous Dirichlet problem for first-order Hamilton--Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $\Omega$ of $\mathbb{R}^n$ 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, …