p-Laplace Label Extension — Misapplied Maths

What is this doing?

The $p$ -Laplace operator is

Δ_{p} u = div (∣\nabla u ∣^{p - 2} \nabla u) .

For $p = 2$ this is the ordinary Laplacian and $Δ_{p} u = 0$ is just Laplace's equation; for $p \neq = 2$ the equation is genuinely nonlinear, with coefficients that depend on the solution's own gradient.

The visualization above samples values from the rose-coloured ground truth at the points you click and then extends those samples to the whole grid by solving $Δ_{p} u = 0$ with the sampled values as Dirichlet data. The blue surface is the extension.

The character of the extension depends strongly on $p$ . Near $p = 2$ the extension is harmonic and uniformly smooth, distributing each sample's influence by averaging. As $p$ drops toward $1$ the equation degenerates and the solver becomes a total-variation flow: large flat regions separated by sharp transitions become energetically cheap, and the extension starts to look piecewise-constant.

In the opposite limit, as $p \to \infty$ , the $p$ -harmonic equation degenerates the other way and the solution converges to the absolutely minimizing Lipschitz extension (AMLE), which is characterised by the infinity-Laplace equation

Δ_{\infty} u = i, j \sum \partial_{i} u \partial_{j} u \partial_{ij} u = 0.

Informally, AMLE is the extension that stretches as little as possible between labelled points — its local Lipschitz constant is minimal on every subdomain. Drag the slider to sweep across these regimes for the same set of labelled points; orbit the camera with the mouse to see the surfaces from different angles.