Higher differentiability of minimizers of convex variational integrals

Abstract

In this paper we consider integral functionals of the formwith convex integrand satisfying (p, q) growth conditions. We prove local higher differentiability results for bounded minimizers of the functional F under dimension-free conditions on the gap between the growth and the coercivity exponents.As a novel feature, the main results are achieved through uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand.

Calderón–Zygmund estimates and non-uniformly elliptic operators

“…Such non-standard growth conditions (in Marcellini's terminology [40]), have been attracting increasing attention, starting by the pioneering papers of Marcellini [40][41][42]. See for instance [5,13,22,36,44,47] and our previous papers [15,16] for the regularity of minima of the functional P.…”

“…Again the counterexamples in [22,23] show that condition (1.13) cannot be dropped from the assumptions. The possibility of relaxing assumption when solutions are bounded is a classical fact, observed for instance in [5,18,13,14,21,37,51], where bounded minima of functional as in (1.6) are considered. Here this fact is also observed at the level of Calderón-Zygmund theory.…”

Higher differentiability of minimizers of variational integrals with Sobolev coefficients

“…A different approach has been introduced in [5] and it is based on establishing higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand. A different approach has been introduced in [5] and it is based on establishing higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand.…”

Regularity results for a new class of functionals with non-standard growth conditions

“…In 1991, P. Marcellini started the study of the regularity of local minimizers under the more flexible (p, q)-growth assumption |z| p f (x, z) L 1 + |z| q , 1 < p < q (1.2) (see [27][28][29]), stimulating the interest for the theory of regularity of integrals with non-standard growth conditions and that, since then, have received many contributions from several authors (see for example [1,5,8,9,[14][15][16]30]). An interesting class of functionals that satisfy non-standard growth conditions are those with ϕ-growth, i.e.…”