Remarks on the Vanishing Viscosity Process of State-Constraint Hamilton–Jacobi Equations

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

“…According to this proposition, the local semiconcavity constant of u is bounded by the inverse of the time that a minimizing curve takes to hit the boundary. Compared with previous results [9] or Corollary 2.1, where the semiconcavity constant is bounded by the inverse of dist(x, ∂Ω), (5) provides a better bound for the semiconcavity constant when x is close to the boundary of the domain. More specifically, when dist(x, ∂Ω) is extremely small, although dist(x, ∂Ω) −1 ≫ 1, the minimizing curves for u(x) may have relatively slow speed and take a long time to hit the boundary, which makes 1/T remain bounded.…”

“…Since the speed of minimizing curves is uniformly bounded, we can then deduce that the semiconcavity constant of u at the point x depends on dist(x, ∂Ω), which we summarize below in Corollary 2.1. (See also [9]) Corollary 2.1. Under the conditions of Proposition 1.1, there exists a constant C > 0 independent of x ∈ Ω so that ∀x ∈ Ω,…”

“…It is not hard to show that the solution u is locally semiconcave. As is proved in [9,Theorem 16], the local semiconcavity constant of u at a specific point x is bounded by the inverse of its distance to the boundary, i.e., dist(x, ∂Ω) −1 . But this does not imply global semiconcavity since dist(x, ∂Ω) −1 blows up near the boundary.…”

“…The following lemma illustrates the relation between the function f and the solution u that they vanish at the same points, which follows from the optimal control formula of the constrained solution. See [9] for proof.…”

“…It is proved in [9] that if f can be extended to a function f ∈ C(R n ) by setting f = 0 in Ω c so that f is semiconcave in R n , then u is globally semiconcave (see [9]). For instance, if f ∈ C 2 c (Ω), then f can be extended to a semiconcave function on R n by setting f = 0 outside Ω.…”