Symmetry breaking for local minimizers of a free discontinuity problem

TL;DR

This paper investigates a free discontinuity problem, revealing that in two dimensions, symmetry breaking leads to complex minimizers with non-striped level sets, contrasting the one-dimensional staircase solutions.

Contribution

It introduces a novel adaptation of the calibration method for free discontinuity problems, demonstrating symmetry breaking in two dimensions.

Findings

Symmetry breaking occurs in two-dimensional minimizers.

Level sets are not simple stripes in 2D minimizers.

The adapted calibration method requires less regularity.

Abstract

We study a functional defined on the class of piecewise constant functions, combining a jump penalization, which discourages discontinuities, with a fidelity term that penalizes deviations from a given linear function, called the forcing term. In one dimension, it is not difficult to see that local minimizers form staircases that approximate the forcing term. Here we show that in two dimensions symmetry breaking occurs, leading to the emergence of exotic minimizers whose level sets are not simple stripes with boundaries orthogonal to the gradient of the forcing term. The proof relies on a suitable adaptation of the calibration method for free discontinuity problems; as a side benefit, our version requires less regularity than the classical one.