Ž . Journal of Mathematical Analysis and Applications 235, 578597 1999 Article ID jmaa.1999.6410, available online at http:www.idealibrary.com on Holder Continuity of Local Minimizers ¨ Giovanni Cupini, Nicola Fusco, and Raffaella Petti Dipartimento di Matematica, ‘‘Ulisse Dini,’’ Viale Morgagni 67  A, 50134 Florence, Italy Submitted by Arrigo Cellina Received March 19, 1999 1. INTRODUCTION In recent years manyresults have appeared concerning the regularity of minimizers of integral functionals of the type F  ;   F x ,  x , D x dx , 1.1 Ž . Ž . Ž . Ž . Ž . H  where F :      N   is an integrand satisfying the growth assump- tion p2 p2 2 2 2 2       z  F x , u , z  L   z 1.2 Ž . Ž . Ž . Ž . with L  1,   0, p  1. Roughly speaking two kinds of results are available. Ž . If no other assumption is made on the integrand F, it is known see 7 Ž . 1, p that condition 1.2 ensures that a W minimizer u is Holder continuous ¨ for some exponent  depending on L, p and N. On the other hand, if F is assumed to be smooth enough, for instance C 2 with respect to z, and satisfies a standard ellipticity assumption of the form N Ž . p2 2 2 2 2 N     DF x , u , z      z      , 1.3 Ž . Ž . Ž . Ý ij i j i , j1 Ž  . one gets that Du is Holder continuous see, e.g., 1, 3, 8, 12 . ¨ If one is interested only into Lipschitz continuity properties of minimiz- ers, the situation is somewhat different. In fact a classical result due to 578 0022-247X99 $30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.