The Regularity of Lagrangians f(a.) = a(z) with Hölder Exponents a(z)

In this paper the regularity of the Lagrangians f(x,t) = a(z) (1 < a < a(z) . a. < +00) is studied. Our main result: If a(z) is Hölder continuous, then the Lagrangian f(x,E)= |.ja(z) is regular. This result gives a negative answer to a conjecture of V. Zhikov.