Proceedings of the Royal Society of Edinburgh, 133A , 1249{1262, 2003 Partial regularity of local minimizers of quasiconvex integrals with sub-quadratic growth Menita Carozza Universit´a del Sannio, Via dei Mulini, 82100 Benevento, Italy Antonia Passarelli di Napoli Dipartimento di Matematica e Applicazioni, Universita Monte S. Angelo, Via Cintia, 80126 Napoli, Italy (MS received 19 July 2002; accepted 28 March 2003) We prove partial regularity for local minimizers of quasiconvex integrals of the form I (v )= R « F (Dv(x)) dx, where the integrand f (¹ ) has sub-quadratic growth, i.e. jF (¹ )j 6 L(1 + j¹ j p ), with 1 <p< 2. A function u 2 W 1;p (« ; R N ) is a W 1;q (« ; R N ) local minimizer of I (v) if there exists a ¯> 0 such that I (u) 6 I (v ) whenever v 2 u + W 1;p 0 (« ; R N ) and kDv ¡ Dukq 6 ¯ . 1. Introduction Let us consider the functional I (v)= Z « F (Dv (x)) dx; where « is a bounded open subset of R n , v is a W 1;p («; R N ) function with p> 1 and F (¹ ): R nN ! R is a C 2 uniformly strict quasiconvex function, i.e. Z « F (¹ + D¿(x)) dx > Z « · F (¹ )+ ¸ (1 + j¹ j 2 + jD¿(x)j 2 ) (p¡2)=2 jD¿(x)j 2 ¸ dx (1.1) for any ¹ 2 R nN and ¿ 2 C 1 0 (« ; R N ). This condition was introduced for the case p > 2 in a paper by Evans (see [5]). He proved that if f satis-es (1.1) and jD 2 F (¹ )j 6 L(1 + j¹ j 2 ) (p¡2)=2 ; (1.2) then a minimizer of I (u) is C 1;¬ on an open subset « 0 » « such that meas(« n « 0 )=0: We refer to a function u 2 W 1;p (« ; R N ) as a minimizer of the functional I if I (u) 6 I (u + ¿) for every function ¿ 2 W 1;p 0 («; R N ). Evans’s result was later generalized in [1], where condition (1.2) is dropped (see also [9]). In 1991, µ Sverak (see [13]) gave an example of a quasiconvex, not convex neither polyconvex, function having polynomial growth with exponent 1 <p< 2. 1249 c ® 2003 The Royal Society of Edinburgh