A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

Let $(X,\omega_0)$ be a compact K\"ahler manifold, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. For the equilibrium envelope $u_0$ associated with $X\setminus U$, the normalized Monge--Amp\`ere measure vanishes on $U$ and equals the background measure $V^{-1}\omega_0^n$ on $X\setminus\overline U$; its remaining component is a singular measure supported on $\partial U$, 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--Amp\`ere energy along the outward parallel family $U_\varepsilon=\{\rho<\varepsilon\}$. The nonlinear telescoping identity gives the mixed Bedford--Taylor boundary traces that sum up to the trace of the current $$\mathcal J_{\mathrm{tot}} =\frac1V\mathrm{d}^c u_0\wedge\sum_{p=0}^{n-1}(p+1)\omega_{u_0}^p\wedge\omega_0^{n-1-p}, \qquad \omega_{u_0}=\omega_0+\mathrm{d}\mathrm{d}^c u_0.$$ 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--Amp\`ere growth.