The paper is devoted to the Lagrange problem in the bounded region for first-order partial differential inclusions (PDIs). For this, using discretisation method and locally adjoint mappings (LAMs), in the form of Euler-Lagrange type inclusions and conjugate boundary conditions, sufficient optimality conditions are obtained. The transition to a continuous problem with PDIs is possible using a specially proved equivalence theorem. To demonstrate this approach, some semilinear problems and polyhedral optimisation with first-order partial differential inclusions are considered. Furthermore, the numerical results also are provided.