Monotonicity of Dirichlet energy along the H-surface flow

Determine whether the Dirichlet energy is monotone along the H-surface flow with Dirichlet boundary condition.

Background

The paper studies the parabolic H-surface flow associated with the prescribed-mean-curvature system. Unlike the harmonic map flow, whose Dirichlet energy is known to be monotone under appropriate conditions, the H-surface flow has a monotone enclosed-volume-corrected energy functional E_H, but this does not by itself imply monotonicity of the Dirichlet energy D(u). Establishing whether D(u(·,t)) is non-increasing is relevant to global existence, energy concentration, and the analysis of weak solutions without the small-energy assumptions used in the paper.

References

However, unlike the situation with the harmonic map flow, it is not clear whether the Dirichlet energy is indeed monotone along the $H$-surface flow, making the study of the latter more subtle.

— Energy convexity and uniformity of the $H$-surface flow in $\mathbb{R}^{3}$ with Dirichlet boundary condition  (2609.30740 - Cheng et al., 25 Sep 2026) in Introduction, paragraph reviewing Chen–Levine (2002) and related work