Solutions of vectorial Hamilton–Jacobi equations and minimizers of nonquasiconvex functionals

We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton–Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.