This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of Legendre-Fenchel transforms the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems and duality theorems are proved. The results obtained are generalized to the multidimensional case with a second order elliptic operator.

This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of Legendre-Fenchel transforms the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems and duality theorems are proved. The results obtained are generalized to the multidimensional case with a second order elliptic operator.