Regularity For Non-Uniformly Elliptic Systems And Application To Some Variational Integrals

. This paper deals with higher integrability for minimizers of some variational integrals whose Euler equation is elliptic but not uniformly elliptic. This setting is also referred to as elliptic equations with p; q-growth conditions, following Marcellini. Higher integrability of minimizers implies the existence of second derivatives. This improves on a result by Acerbi and Fusco concerning the estimate of the (possibly) singular set of minimizers. 0. Introduction Let\Omega be a bounded open set of R n , n 2, u be a (possibly) vector-valued function, u :\Omega ! R N ; N 1, F be a continuous function, F : R nN ! R; we consider the integral I(u) = Z \Omega F (Du(x)) dx ; (0.1) where jF (¸)j c (1 + j¸j p ) ; (0.2) u 2 W 1;p ; 2 p. Regularity of minimizers has been widely studied when m (1 + j¸j p\Gamma2 )jj 2 DDF (¸) ; 0 ! m ; (0.3) jDDF (¸)j c (1 + j¸j p\Gamma2 ); (0.4) see [24], [14], [16], [17] (and [10], [12], [18], [11], [20], where (0.3) is weakened..