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State-constraint static Hamilton-Jacobi equations in nested domains

Yeoneung Kim, Hung V. Tran, Son N. T. Tu

arXiv:1906.11222v2math.AP

TL;DR

The paper asks how quickly state-constraint solutions on nested domains converge to the solution on their union. It proves rates for two prototype families, with optimality established in several cases. The results include a polynomial estimate for general nonconvex Hamiltonians in one family and exponential rates in specified cases.

  • Problem

    Although prior work on state-constraint problems in nested domains provided qualitative results, the paper asks how fast the domainwise solutions converge to the solution on the union.

    Source: §1 · p. 1

  • Method

    The paper analyzes state-constraint solutions on two prototype families of nested domains using viscosity-solution estimates and, for convex Hamiltonians, optimal control theory.

    Source: §1 · p. 1

  • Results

    For prototype (P1), 0 ≤ u_k(x) − u(x) ≤ C(1 + R^2)/k^2 for |x| ≤ R, while optimal exponential rates are established in specified cases.

    Source: §1.2 · p. 3

  • Takeaways & Limitations

    The results provide quantitative convergence rates for two nested-domain prototypes, with optimality proved in several cases.

    Source: §1.2 · p. 3

    The stated estimate is restricted to bounded locations: at points on ∂Ω_k, examples can have |u_k(x) − u(x)| = 1.

    Source: §1.2 · p. 3

Abstract

from arXiv · show

We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}_{k \in \mathbb{N}}$ in $\mathbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \mathbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega = \bigcup_{k \in \mathbb{N}} \Omega_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.

1. Introduction

The paper studies convergence of state-constraint solutions on nested domains and gives quantitative rates for two prototype domain families. Under stated assumptions, several rates are shown to be optimal, and the paper also discusses examples and extensions.

1.1. Assumptions.

  • The assumptions include a convexity condition on H in momentum, and the paper states that H1 is necessary for asking a meaningful convergence-rate question.The paper is organized around well-posedness results, proofs for the rate theorems, and examples and discussion.Source: §1.1 · p. 2, §1.3 · p. 4

1.2. Main results.

  • The paper frames its contribution against prior nested-domain results that were qualitative and required approximations, and presents its rates as initial quantitative results.The discussion says the results concern two prototype domain families and may extend to more general domains under appropriate conditions.Source: §1.2 · pp. 3–4
  • For prototype (P1), the general nonconvex estimate is 0 ≤ u_k(x) − u(x) ≤ C(1 + R^2)/k^2 for |x| ≤ R, with C depending only on H.The estimate applies for fixed R and k ≥ R; the cited result also gives u(x) ≤ u_k(x).Source: §1.2 · p. 3
  • For (P1), the paper also establishes optimal exponential convergence for H(x, p) = a(x)K(p) under its stated hypotheses, and for autonomous coercive locally Lipschitz H(p).When a(x) is a positive constant, the condition K(0) = 0 can be removed.Source: §1.2 · p. 3
  • For convex Hamiltonians, the analysis uses optimal control theory; a bounded-velocity minimizer is a key element in proving the stated convergence result.The paper says the velocity bound depends on the growth of H, not its smoothness.Source: §1.2 · p. 4
  • For prototype (P2), the paper proves an optimal convergence rate for a quite general class of Hamiltonians.The supplied theorem passages do not state the rate's formula.Source: §1.2 · p. 4

2. Preliminaries

The preliminaries define state-constraint viscosity solutions and establish conditions for their existence and uniqueness. The state constraint also imposes boundary behavior, and comparison supports uniqueness.

2. Preliminaries

  • BUC(Ω; R) denotes the bounded uniformly continuous functions defined on Ω.Source: §2 · p. 5
  • A state-constraint solution is a viscosity subsolution in the domain and a viscosity supersolution on its closure.The paper gives these conditions in the viscosity sense.Source: §2 · p. 5
  • For a smooth boundary and a C1 solution, the state constraint implicitly imposes a boundary condition involving the outward normal.The passage points to a normal-derivative condition but does not provide its formula here.Source: §2 · p. 5
  • Under (H1), (H2), and (H4), a state-constraint solution exists in C(Ω) ∩ W1,∞(Ω).The construction uses Perron’s method, with coercivity providing an a priori estimate.Source: §2 · p. 5
  • Under (A), comparison holds when either stated condition applies, including (H3b) when the supersolution is Lipschitz.Comparison yields uniqueness when applicable; the paper identifies the unique solution as the maximal viscosity subsolution.Source: §2 · pp. 6–7

3. A rate of convergence for general Hamiltonians in unbounded domain

For the first prototype, the paper establishes local uniform convergence on R^n and a rate for general Hamiltonians. A doubling-variables argument gives an initial O(1/k) bound, which monotonicity improves; optimality remains unresolved in the general nonconvex setting.

3. A rate of convergence for general Hamiltonians in unbounded domain

  • For the first prototype, the argument assumes (H1), (H2), (H3c), and (H4), with u_k the unique solution on B(0, k).The solutions satisfy the state-constraint problem on each ball.Source: §3 · p. 7
  • The solutions converge locally uniformly on R^n to a viscosity solution of the limiting problem.The limit is identified using stability of viscosity solutions and uniqueness.Source: §3 · p. 7
  • The doubling-variables argument gives an O(1/k) rate on each fixed ball B(0, R).The passage describes this as a typical rate from the doubling-variables method.Source: §3 · pp. 7–8
  • Monotonicity of the solutions bounds the maximizing point |y_k| by a constant, enabling a further improvement over the initial rate.The earlier bound was |y_k| ≤ k; the bootstrap uses Φ_k(x_k, y_k) ≥ Φ_k(0, 0) together with u_k ≥ u.Source: §3 · pp. 8–9
  • Whether the convergence rate is optimal in the general nonconvex setting remains unknown.The paper notes that the doubling-variables rate appears to be the best this method can obtain, since |y_k| is unlikely to vanish.Source: §3 · p. 9

4. An optimal rate for a class of nonconvex Hamiltonians on unbounded domain

For Hamiltonians of the form H(x, p) = a(x)K(p), with K(0) = 0 and 0 < α ≤ a(x) ≤ β, the paper proves an optimal exponential convergence rate, including for a class of possibly nonconvex Hamiltonians. The rate depends on regularity: Hölder continuity around 0 can yield slower convergence, while the K(p) + V(x) form is more complicated.

4. An optimal rate for a class of nonconvex Hamiltonians on unbounded domain

  • O(e^-Ck) is an optimal convergence rate for H(x, p) = a(x)K(p), with K(0) = 0 and 0 < α ≤ a(x) ≤ β.The result applies to a class of possibly nonconvex Hamiltonians.Source: §4 · p. 9
  • A symmetric Hamiltonian ˜H with ˜H(0) = 0 and ˜H ≤ H is used to construct a supersolution for the state-constraint problem.The construction is described as a brief idea for the proof.Source: §4 · p. 9
  • When H is only Hölder continuous around 0, convergence is slower, and its optimal rate can be arbitrarily slow as γ → 0+.The paper identifies condition (H3c) as essential for this section.Source: §4 · pp. 11–12

5. An optimal rate for convex Hamiltonians

For convex Hamiltonians, the paper uses an optimal-control representation and proves an exponential convergence rate using minimizing paths with bounded velocity. It also gives examples of exponential convergence, including one where the constrained optimal path eventually remains on the boundary.

5. An optimal rate for convex Hamiltonians

  • Theorem 5.7 gives a minimizer η for each x with J[x, η] = u(x) and a uniform weighted L2 bound on its velocity.The bound is ∥e^-s/2 η̇(s)∥L2(0,∞) ≤ C4, where C4 depends only on CH, A, and B.Source: §5 · p. 14
  • The proof of the optimal rate uses minimizing paths with bounded velocity in the optimal-control representation.The velocity bound depends only on CH, A, and B, and the convergence estimate does not depend on Lagrangian regularity.Source: §5 · pp. 14, 16

5.1. Examples with exponential rate of convergence

  • In Proposition 5.10, uk converges locally uniformly to 0, but uk(k) = 1 for every k, so convergence fails at the moving endpoint.The example uses uk(x) = e^(x-k) on [−k, k].Source: §5.1 · p. 16

5.2. Optimal control formulations

  • In Proposition 5.11, the solution on [−k, k] converges to u(x) = e^-|x| at an exponential rate, with convergence throughout the domain.The result passage states that convergence holds everywhere in [−k, k].Source: §5.2 · pp. 17–18
  • For x0 ≥ 0, the constrained optimal control moves at rate 1 until reaching k, then stays at 0; for x0 < 0, it analogously moves toward −k.The optimal path stays on the boundary ±k for all t ≥ k − |x0|.Source: §5.2 · p. 18

6. The case of bounded domain

In the bounded-domain case, the paper establishes locally uniform convergence of the state-constraint solutions to a limiting solution. Its proof of the rate result uses the star-shaped and scaling properties of the domains, and the discussion notes a relaxation of prototype condition (P2).

6. The case of bounded domain

  • For each k, the unique solution uk converges locally uniformly to a limiting u that satisfies the state-constraint equation and retains the stated bounds.Source: §6 · p. 18
  • The proof of Theorem 1.5 relies on the star-shaped and scaling properties of the nested domains.Source: §6 · p. 19
  • The paper states that prototype condition (P2) can be relaxed.Source: §6 · p. 19

7. Discussions

The discussion contrasts convergence rates across Hamiltonian settings, relaxes a prototype domain condition, and identifies open rate questions. Its examples include an exponential rate for an additive Hamiltonian and a polynomial rate whose exponent can vary.

Examples and rates

  • For the first-prototype Hamiltonian H(x, p) = a(x)K(p), imposing (H1) gives an exponential convergence rate; without (H1), an example has a polynomial rate whose power can be increased or decreased arbitrarily.Source: §7 · p. 20
  • The example’s limit 0 is not the only solution of (3.3): another solution is excluded from BUC(R).The displayed other solution is u(x) = (m−1)(1 + |x|)^m.Source: §7 · p. 20

Relaxing (P1)

  • Condition (P1) can be relaxed to require bounded Ωk containing B(0, k), with comparison for the state-constraint problem on each Ωk.The discussion says this relaxation follows from the proofs of the main results.Source: §7 · pp. 20–21

Open questions

  • Open questions ask for the optimal rate in the general nonconvex (P1) case and whether K(p) + V(x) always yields an exponential rate under the stated nonconvex assumptions.The second question assumes coercive, Lipschitz, nonconvex K and bounded uniformly continuous V.Source: §7 · p. 21

8. Appendix

The appendix supplies proof ingredients for the convergence analysis, including bounds on minimizing trajectories and an existence argument that removes smoothness of the Lagrangian under (L3). It also uses smoothing and stability to recover solutions for the original Hamiltonian.

8.1. Proofs of some lemmas.

  • The argument establishes J[x, η] = u(x) by showing J[x, η] ≤ u(x) and using the minimizing construction.The appendix constructs a minimizing sequence and obtains a limiting admissible path.Source: §8.1 · pp. 21–22
  • The minimizing trajectory η has velocity bounded by C5, with the bound depending only on CH, A, and B.The proof states this dependence explicitly.Source: §8.1 · p. 23

8.2. Existence of minimizers in the general case.

  • Under (L3), the proof removes the smoothness requirement on L in Theorem 5.7 and obtains a minimizer using convexity and weak lower semicontinuity.The functional on L2(µ) is convex and lower semicontinuous, and the weak convergence argument yields the minimizer.Source: §8.2 · p. 23
  • Smoothing L to Lε yields solutions uε that converge locally uniformly to u as ε tends to zero.The passage states that Hε converges locally uniformly to H and invokes stability of viscosity solutions.Source: §8.2 · p. 23
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