Global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints

We focus on the global semiconcavity of solutions to first-order Hamilton–Jacobi equations with state constraints, especially for the Hamiltonian $H(x, \beta ):=|\beta |^{p}-f(x)$ with $p \in (1, 2]$ . We first show that the solution is locally semiconcave and that the semiconcavity constant at each point depends on the first time a corresponding minimizing curve originating from this point hits the boundary. Then, with appropriate conditions on Df, we prove that for any such minimizing curve, the time it takes to hit the boundary of the domain is +∞, and as a consequence, the solution is globally semiconcave. Moreover, the condition on Df is essentially optimal with examples in one-dimensional space. The proofs employ the Euler–Lagrange equations and techniques in weak KAM theory.