2026/07/19 by Ziyu Li
#math.CV #math.AP #math.DG
Let (X,ω0) be a compact Kähler manifold, and let U\Subset X have C3,1 uniformly strongly pseudoconvex boundary. For the equilibrium envelope u0 associated with X∖ U, the normalized Monge--Ampère measure vanishes on U and equals the background measure V-1ω0n on X∖ U; its remaining component is a singular measure supported on ∂ U, where V=∫Xω0n. 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 Uε=\ρ<ε\. The nonlinear telescoping identity gives the mixed Bedford--Taylor boundary traces that sum up to the trace of the current \mathcal Jtot =\frac1Vdc u0\wedge∑p=0n-1(p+1)ωu0p\wedgeω0n-1-p, ωu0=ω0+ddc u0. 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.