A Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth

TL;DR

This paper provides a precise condition linking the decay of total variation to the continuity of local minima of parabolic variational integrals with linear growth, including solutions to the parabolic 1-Laplacian.

Contribution

It establishes a necessary and sufficient condition for the continuity of minimizers and solutions to the parabolic 1-Laplacian based on total variation decay.

Findings

Continuity of minimizers characterized by total variation decay.

Condition applies to solutions of the parabolic 1-Laplacian.

Provides a unified criterion for regularity in variational problems.

Abstract

For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of the total variation of $u$ about $(x_o,t_o)$ (see (1.4) below). These minimizers arise also as {proper} solutions to the parabolic $1$-laplacian equation. Hence, the continuity condition continues to hold for such solutions (\S\ 3).