On $\infty$-Ground States in the Plane

TL;DR

This paper investigates the properties of $ abla$-ground states in convex planar domains, characterizing non-$ abla$-Laplace points in polygons as lying on specific curves that act as streamlines, with continuous gradients outside these curves.

Contribution

It provides a geometric characterization of points where $ abla$-ground states do not satisfy the $ abla$-Laplace Equation in convex polygons, identifying attracting streamlines.

Findings

Points not satisfying the $ abla$-Laplace Equation lie on specific curves.

Gradient is continuous outside these curves.

Streamlines do not intersect outside these curves.

Abstract

We study $\infty$-Ground states in convex domains in the plane. In a polygon, the points where an $\infty$-Ground state does not satisfy the $\infty$-Laplace Equation are characterized: they are restricted to lie on specific curves, which are acting as attracting (fictitious) streamlines. The gradient is continuous outside these curves and no streamlines can meet there.