Jensen Deficits for Inhomogeneous Monge-Amp\`ere Dirichlet Problems

We develop an inhomogeneous form of Edwards' Jensen-measure duality for Perron envelopes constrained by Monge--Amp\`ere lower bounds. The admissible subsolution families are convex but not cones; nevertheless, the dual measures remain the homogeneous Jensen measures, and the right-hand side enters through a scalar Jensen deficit \[ B_{\mathcal{A}}(x,\mu) = \inf_{u\in\mathcal{A}} \left(\int_{\partial\Omega}u\,d\mu-u(x)\right). \] Under natural structural hypotheses we prove a boundary dual formula \[ \sup\{u(x):u\in\mathcal{A},\ u\leq\varphi\text{ on }E\} = \inf_{\mu\in J_x^\partial} \left( \int_{\partial\Omega}\varphi\,d\mu - B_{\mathcal{A}}(x,\mu) \right). \] We apply the theorem to real Alexandrov subsolutions and to complex Bedford--Taylor plurisubharmonic subsolutions with continuous density. In one real dimension the deficit is the Green-potential correction; in higher dimensions it has intrinsic stress and current interpretations. On B-regular domains, a bounded Bedford--Taylor approximation theorem identifies bounded and continuous competitors and yields a duality proof of continuity for the corresponding Dirichlet solution. Finally, for smooth strictly elliptic solutions, optimal Jensen measures are the harmonic measures of the linearized Monge--Amp\`ere operators, equivalently the boundary derivatives of the nonlinear solution map.