An energy-decreasing rearrangement of level sets preserving the domain and volume constraints

Let Ω⊂R2 be a bounded and regular domain, u∈C3(Ω) and V⊂Ω a domain where the subset K0 of points where the curvature of the t-level sets of u is zero admits a regular t-parameterization. We exhibit a local correction of u in a neighborhood of a particular point x∗∈K0⊂V such that the volume ∫f(u) is preserved and the Dirichlet integral ∫2|∇u| decreases. Consequently, a certain monotonic property is deduced for constrained minimizers in H1(Ω). Our result can be applied to classical variational and free-boundary problems.