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Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times

Siyao Zhu, Hongguang Wu, Xiaojun Cui

Abstract

In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex.

Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times

Abstract

In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex.
Paper Structure (15 sections, 27 theorems, 85 equations, 2 figures)

This paper contains 15 sections, 27 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Construction of global viscosity solutions to equation (\ref{['E1']})
  4. The construction of $u^{\pm}$
  5. A priori estimate of $u^+$
  6. $u^+$ is global viscosity solution to equation \ref{['E1']}
  7. The level sets of viscosity solution of Lorentzian eikonal equation
  8. Equivalent characterization of time orientation of viscosity solutions
  9. The level sets of $u$ are partial Cauchy surfaces
  10. A variational representation of $u$
  11. The existence of minimizer to the variational problem
  12. Variational representation and forward calibrated curve
  13. Local semiconcavity of $u$
  14. Weak KAM properties of $u$
  15. Characterization of $C_u$

Key Result

Theorem 1

If $(M, g)$ is a globally hyperbolic space-time, then $\mathcal{S}(M)\neq \emptyset$. In other words, the eikonal equation admits at least one globally defined locally Lipschitz viscosity solution.

Figures (2)

  • Figure 1: A space-time admits a viscosity solution $u(x,y)=x$. Obviously, any Cauchy surface $\Gamma$ (the blue curve) must connect $(0, -1)$ and $(0, 1)$. The level set $u_a$ (the red segment) is not a Cauchy surface except $a=0$.
  • Figure 2: A globally defined viscosity solution to equation (\ref{['E1']}) on the 2-dimensional Minkowski space-time.

Theorems & Definitions (59)

  • definition 1: Beem 1996BernalCQG2007
  • definition 2: CannarsaPNDE2004
  • definition 3
  • definition 4: CuiJMP2014
  • definition 5
  • definition 6: Forward calibrated curve
  • definition 7: Partial Cauchy Surface
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • ...and 49 more