Work

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

Year: 2025

Type: article

Abstract: 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.

Source: Advances in Continuous and Discrete Models

DOI: 10.1186/s13662-025-03964-2

Author: Yuxi Han

Institution: Purdue University West Lafayette

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $1,890

Funder: National Science Foundation