Existence of planar curves minimizing length and curvature

TL;DR

This paper investigates the existence of curves minimizing a combined length and curvature functional, revealing conditions under which minimizers exist or do not, and characterizing their geometric features such as cusps and angles.

Contribution

It establishes existence results for minimizers of a length-curvature functional under various boundary conditions, including cases with and without directional orientation.

Findings

Non-existence of minimizers with oriented directions due to angle formation.

Existence of minimizers for the reparameterized functional with non-oriented directions.

Minimizers can have cusps but not angles, depending on boundary conditions.

Abstract

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the "time-reparameterized" functional $$\int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.