A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

Let $(X,{\omega }_0)$ be a compact Kähler manifold, and let $U⋐X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. For the equilibrium envelope $u_0$ associated with $X\setminus U$, the normalized Monge–Ampère measure vanishes on $U$ and equals the background measure $V^{-1}{\omega }_0^n$ on $X\setminus \bar{U}$; its remaining component is a singular measure supported on $\mathrm{\partial }U$, where $V={\int }_X{\omega }_0^n$. We identify this boundary component as the negative outward Anzellotti trace of a divergence-measure flux current. We then prove a one-sided Hadamard formula for the normalized Monge–Ampère energy along the outward parallel family $U_{\epsilon }=\{\rho <\epsilon \}$. The nonlinear telescoping identity gives the mixed Bedford–Taylor boundary traces that sum up to the trace of the current $${\mathcal{J}}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{1}{V}{\mathrm{d}}^c{u}_0\wedge \sum _{p=0}^{n-1}(p+1){\omega }_{u_0}^p\wedge {\omega }_0^{n-1-p},{\omega }_{u_0}={\omega }_0+\mathrm{d}{\mathrm{d}}^c{u}_0\mathrm{.}$$ These results provide a local weak formulation of boundary flux and normal variation for regular interface problems related to Darcy/Hele–Shaw type problems and Monge–Ampère growth.