Semiconcavity estimates for viscous Hamilton–Jacobi equations

Strömberg, Thomas

doi:10.1007/s00013-010-0132-2

Cited by 3 publications

(1 citation statement)

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“…For local semiconcavity of Hamilton-Jacobi equations with state constraints, see [3,20]. There are also various results regarding semiconcavity of different types of equations, for instance, [1,2,6,17,16,20,22]. Second-order Hamilton-Jacobi equations with state constraints are studied in various work, for instance, [14,18,19].…”

Section: Introductionmentioning

confidence: 99%

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

Han¹

2022

Preprint

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We focus on the global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints, especially for the Hamiltonian H(x, β) := |β| p − f (x) with p ∈ (1, 2]. We first show that the solution is locally semiconcave, and the semiconcavity constant at each point depends on the first time a corresponding minimizing curve emanating 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.

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