The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x)

In this paper the regularity of the Lagrangiansf(x, ξ)=|ξ|α(x)(1<α1≤α(x)≤α2< +∞) is studied. Our main result: Ifα(x) is Holder continuous, then the Lagrangianf(x, ξ)=f(x, ξ)=|ξ|α(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.