  The paper is devoted to an analysis of optimality conditions for nonsmooth
multidimensional problems of the calculus of variations with various types of
constraints, such as additional constraints at the boundary and isoperimetric
constraints. To derive optimality conditions, we study generalised concepts of
differentiability of nonsmooth functions called codifferentiability and
quasidifferentiability. Under some natural and easily verifiable assumptions we
prove that a nonsmooth integral functional defined on the Sobolev space is
continuously codifferentiable and compute its codifferential and
quasidifferential. Then we apply general optimality conditions for nonsmooth
optimisation problems in Banach spaces to obtain optimality conditions for
nonsmooth problems of the calculus of variations. Through a series of simple
examples we demonstrate that our optimality conditions are sometimes better
than existing ones in terms of various subdifferentials, in the sense that our
optimality conditions can detect the non-optimality of a given point, when
subdifferential-based optimality conditions fail to disqualify this point as
non-optimal.
