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Stability of metric viscosity solutions under Hausdorff convergence

Shimpei Makida, Atsushi Nakayasu

Abstract

This study investigated the stability of Hamilton--Jacobi equation on general metric spaces with a perturbation in some whole space. This type of stability appears in the domain perturbation problem. We find that the stability holds when the set converges in the Hausdorff sense and when the metric converges in some uniform sense. Examples of the perturbed space satisfying these assumptions include network approximation of self-similar sets such as the Sierpiński gasket, junction of shrinking tubes, and lattice lines with the Manhattan distance. We also give supplemental results on time-dependent or noncompact case. Stability can be achieved when the class of test function of metric viscosity solutions is reduced to the squared distance functions, whose proof is also given.

Stability of metric viscosity solutions under Hausdorff convergence

Abstract

This study investigated the stability of Hamilton--Jacobi equation on general metric spaces with a perturbation in some whole space. This type of stability appears in the domain perturbation problem. We find that the stability holds when the set converges in the Hausdorff sense and when the metric converges in some uniform sense. Examples of the perturbed space satisfying these assumptions include network approximation of self-similar sets such as the Sierpiński gasket, junction of shrinking tubes, and lattice lines with the Manhattan distance. We also give supplemental results on time-dependent or noncompact case. Stability can be achieved when the class of test function of metric viscosity solutions is reduced to the squared distance functions, whose proof is also given.
Paper Structure (12 sections, 19 theorems, 105 equations)

This paper contains 12 sections, 19 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Metric viscosity solutions
  3. Main results on stability
  4. Example: Self-similar sets
  5. Iterated function systems
  6. The Vicsek fractal
  7. The Sierpiński gasket
  8. On the Koch curve
  9. Example: Junction of shrinking tubes
  10. Example: Lattice lines and the Manhattan distance
  11. Remarks on time-dependent case
  12. Remarks on noncompact case

Key Result

Lemma 2.1

Let $u, v, w$ be locally Lipschitz continuous functions.

Theorems & Definitions (38)

  • Remark 1.1
  • Lemma 2.1: Maximum principle
  • Lemma 2.2: Squared distance function
  • Definition 2.3: Metric viscosity solutions
  • Theorem 2.4: Comparison principle
  • Theorem 2.5: Unique existence
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Theorem 3.1: Stability
  • ...and 28 more