Short-time positivity preservation for the p-heat flow

Determine whether the p-heat flow, defined as the gradient flow of the p-Cheeger energy in the L² space, preserves positivity for any sufficiently small positive time.

Background

The paper defines the p-heat flow as the L²-gradient flow of the relaxed p-Cheeger energy for a negative exponent p. Although the flow satisfies a comparison principle, constant functions are generally not in the Sobolev class and therefore are not necessarily fixed points. Consequently, positivity preservation is not established in general, and the authors explicitly leave its validity even for short times unresolved.

References

This has the consequence that, in general, the $p$-heat flow does not preserve positivity; it is not even clear if positivity is preserved for any small time.

— Causal Sobolev spaces and gradient flows  (2610.21775 - Braun et al., 15 Oct 2026) in Section 1, Introduction, subsection “Contributions”