Metric-Constrained Eikonal Equation

• The metric-constrained Eikonal equation is a first-order PDE that models distance functions on various metric spaces using local cost functions.
• It unifies frameworks such as optimal control, viscosity solutions, and geometric analysis to address smooth, singular, and discrete settings.
• Key numerical methods include anisotropic fast marching and neural PDE solvers, providing robust computation in complex geometric contexts.

A metric-constrained Eikonal equation is a first-order partial differential equation that governs distance functions and related value functions on metric measure spaces, Riemannian manifolds, and more general (possibly singular or non-smooth) metric spaces where the geometry is determined by a possibly discontinuous, unbounded, or only integrable metric or cost function. The “metric constraint” refers to the requirement that the modulus of the generalized gradient matches the local metric, which may depend intricately on position and possibly only be $p$-integrable, introducing substantial analytic and geometric complexity. The equation and its solution frameworks unify optimal control, viscosity solutions, and geometric analysis in both smooth and singular settings.

1. Formulation and Generalized Metrics

The canonical metric-constrained Eikonal equation takes the form

$|\nabla u|(x) = f(x), \quad x\in\Omega,$

where $f\colon\Omega\to(0,\infty]$ encodes the local metric or running cost, possibly only Borel-measurable, discontinuous, unbounded, or $L^p$-integrable. The equation is supplemented with Dirichlet boundary data $u=g$ on $\partial\Omega$.

For Riemannian manifolds, $f(x)\equiv 1$ and $$|\nabla u|(x) = \sqrt{g^{ij}(x)\partial_i u \partial_j u}$$, so the equation reduces to

$$\sqrt{g^{ij}(x)\partial_i u(x)\partial_j u(x)} = 1,$$

yielding geodesic distance from the source as its unique viscosity/metric solution. In arbitrary metric spaces (possibly fractal, graph-based, or lacking local coordinates), $|\nabla u|$ is replaced by suitable metric slope, local Lipschitz constant, or other variants reflecting the specific structure or lack thereof (Liu et al., 2023, Liu et al., 2020, Lê et al., 2024, Camilli et al., 2014).

Classically, for $|\nabla u|(x) = f(x), \quad x\in\Omega,$0 and $|\nabla u|(x) = f(x), \quad x\in\Omega,$1, standard viscosity theory yields a unique continuous solution, but discontinuous or singular $|\nabla u|(x) = f(x), \quad x\in\Omega,$2 fall outside this regime.

2. Metric Reformation and Path Integrals

The optimal control representation leads to the introduction of an “optical distance”

$|\nabla u|(x) = f(x), \quad x\in\Omega,$3

which generalizes the geodesic metric to arbitrary $|\nabla u|(x) = f(x), \quad x\in\Omega,$4. This metric $|\nabla u|(x) = f(x), \quad x\in\Omega,$5 encodes the minimal “cost” to travel from $|\nabla u|(x) = f(x), \quad x\in\Omega,$6 to $|\nabla u|(x) = f(x), \quad x\in\Omega,$7 with local running cost $|\nabla u|(x) = f(x), \quad x\in\Omega,$8. Under the nondegeneracy $|\nabla u|(x) = f(x), \quad x\in\Omega,$9, $f\colon\Omega\to(0,\infty]$0 is indeed a metric, possibly taking $f\colon\Omega\to(0,\infty]$1, and satisfies monotonicity with the background metric: $f\colon\Omega\to(0,\infty]$2 for all $f\colon\Omega\to(0,\infty]$3 (Liu et al., 2023).

The metric-constrained Eikonal equation is then reformulated as

$f\colon\Omega\to(0,\infty]$4

where $f\colon\Omega\to(0,\infty]$5 is the slope in the $f\colon\Omega\to(0,\infty]$6 metric, and the boundary condition becomes $f\colon\Omega\to(0,\infty]$7 on $f\colon\Omega\to(0,\infty]$8. The unique solution is given by the Hopf-Lax type formula

$f\colon\Omega\to(0,\infty]$9

Pathwise integrability of $L^p$0 along curves is essential for the finiteness and well-posedness of $L^p$1.

3. Solution Notions: Monge, Viscosity, and Metric Solutions

Three principal frameworks for defining a solution are:

• Monge Solutions: $L^p$2 is locally bounded and satisfies, at each $L^p$3,

$L^p$4

interpreted as equality for solutions, $L^p$5 for subsolutions, and $L^p$6 for supersolutions (Liu et al., 2023).

• Metric/Viscosity Solutions: The viscosity concept is extended to metric spaces either via test functions (using local semiconcavity) or the use of metric slopes/gradient modulus, as

$L^p$7

This approach is robust and coincides with the Monge definition in length spaces with $L^p$8 continuous (Liu et al., 2020).

• Global Slope and Non-Length Spaces: In fully general metric spaces (not necessarily length spaces), global slope operators $L^p$9 are introduced, resulting in equations of the form $u=g$0 and solution theory based on pointwise and viscosity principles not reliant on absolute continuity or geodesicity (Lê et al., 2024).

These frameworks are demonstrated to be equivalent under mild hypotheses, and all ensure that the solution is locally Lipschitz in the appropriate metric (Liu et al., 2023, Liu et al., 2020).

4. Existence, Uniqueness, and Regularity

Existence and uniqueness are furnished by dynamic programming/optimal control principles, leveraging the Hopf-Lax formula and comparison theorems. Existence requires the possibility, for all $u=g$1, of finding $u=g$2-integrable curves joining $u=g$3 to $u=g$4; in measure spaces this is ensured by $u=g$5 for $u=g$6 (homogeneous dimension of the underlying space), or $u=g$7 (Liu et al., 2023).

Comparison is established using Ekeland-type variational principles and local topology equivalence of $u=g$8 and $u=g$9, formalized as

$\partial\Omega$0

and further substantiated via explicit modulus-of-curve-family estimates (Liu et al., 2023).

Regularity results include:

• If $\partial\Omega$1, $\partial\Omega$2, then $\partial\Omega$3 is $\partial\Omega$4-Hölder continuous in the original metric $\partial\Omega$5.
• If $\partial\Omega$6, $\partial\Omega$7 is Lipschitz in $\partial\Omega$8.
• For continuous $\partial\Omega$9, these estimates coincide with classical Morrey–Sobolev embeddings (Liu et al., 2023).
• Solutions (in all senses) enforce equality of upper and lower slopes, playing the role of a semi-concavity condition (Liu et al., 2020).

5. Extensions to Singular, Fractal, and Discrete Spaces

Metric-constrained Eikonal theory extends beyond smooth manifolds to:

• Fractals: The Sierpinski gasket admits well-posed Eikonal equations via graph limit and metric viscosity approaches, with control-formula characterizations and convergence proofs from discrete to continuous (Camilli et al., 2014).
• Graphs and General Metric Spaces: Using either local slope or global slope (semigroup) operators, Eikonal-type equations admit existence, uniqueness, and explicit solution formulas even in the absence of a length structure, crucial for analysis on networks/Markov chains (Lê et al., 2024).
• Spaces with Poincaré Inequalities: In metric measure spaces supporting $f(x)\equiv 1$0- or $f(x)\equiv 1$1-Poincaré inequalities with doubling measures, sharp regularity and estimate results hold (e.g., Hölder continuity of solutions) (Liu et al., 2023).

The metric-constrained approach thus unifies the treatment of Eikonal equations in diverse geometric contexts.

6. Numerical Methods and Computational Implications

Efficient computation of metric-constrained Eikonal solutions is critical for geometric analysis, data science, and applications in Riemannian statistics and optimal transport. Key methods include:

• Anisotropic Fast Marching: For Riemannian metrics, lattice-basis reduction and acute mesh constructions are used to enable fast-marching algorithms with $f(x)\equiv 1$2 complexity, robust to strong anisotropy and grid orientation (Mirebeau, 2012, Mirebeau, 2012).
• Gridless and Neural PDE Solvers: Physics-informed neural networks can be trained to minimize the PDE residual for $f(x)\equiv 1$3, enabling differentiable, mesh-free computation of distance functions and geodesic flows (Kelshaw et al., 2024).
• Backtracking for Geodesics: Once the solution $f(x)\equiv 1$4 is obtained, geodesics are recovered by integrating $f(x)\equiv 1$5, yielding globally length-minimizing paths (Kelshaw et al., 2024).
• Control Problems and Assignment: Fast solution of Fréchet means and $f(x)\equiv 1$6-means clustering on manifolds is enabled, utilizing precomputed metric-constrained distance functions (Kelshaw et al., 2024).

7. Applications and Impact

The metric-constrained Eikonal framework has broad impact:

• Geometric Optics and Physics: Determines wavefront propagation in media with variable speed/cost including non-smooth, anisotropic, or discontinuous settings; crucial for general relativity, geometric optics, and inverse problems (Osetrin et al., 2017).
• Analysis on Singular Spaces: Enables analysis, existence, and regularity theory for geodesic distances on fractals and graphs, with applications in network analysis and random walks (Camilli et al., 2014, Lê et al., 2024).
• Statistical and Data Sciences: Permits global computation of Fréchet means, intrinsic clustering, and geodesic paths on statistical manifolds and data-embedded Riemannian geometries (Kelshaw et al., 2024).
• Mathematical Foundations: Provides equivalence of notions (Monge, viscosity, metric) and stability/robustness results necessary for theoretical and computational PDE analysis (Liu et al., 2020, Liu et al., 2023).

The unifying perspective introduced by the metric-constrained Eikonal equation—incorporating singular structures, generalized cost functions, and robust variational and comparison principles—anchors a modern approach to Hamilton–Jacobi theory in metric spaces.