A new sufficiency theorem for strong minima in the calculus of variations

Sufficiency for strong local optimality in the calculus of variations involves, in the classical theory, the strengthened condition of Weierstrass. A proof of sufficiency for strong minima, modifying this condition under certain uniform continuity assumptions on the functions delimiting the problem, is presented. The proof is direct in nature as it makes no use of fields, Hamilton-Jacobi theory, Riccati equations or conjugate points. Some examples illustrate clear advantages of the new sufficient condition over the classical one.