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.
We establish local higher integrability and differentiability results for minimizers of variational integrals! R N satisfying a Dirichlet boundary condition. The integrands F are assumed to be autonomous, convex and of ( p, q) growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range 1 < p q < p ⇤ , where p ⇤ denotes the Sobolev conjugate exponent of p.
Consider the Hardy-Littlewood maximal operatorIt is known that M applied to f twice is pointwise comparable to the maximal operator M L log L f , defined by replacing the meanwhereIn this paper we prove that, more generally, if Φ(t) and Ψ(t) are two Young functions, there exists a third function Θ(t), whose explicit form is given as a function of Φ(t) and Ψ(t), such that the composition M Ψ • M Φ is pointwise comparable to M Θ . Through the paper, given an Orlicz function A(t), by M A f we meanwhere ||f || A,Q = inf λ > 0 :
We prove partial regularity for minimisers of quasiconvex integrals of the form ∫Ωf(Du(x))dx. More precisely, we consider an integrandf(ξ) having subquadratic growth, i.e. |f(ξ)|≦L(1+|ξ|p) withp< 2. The case of a general integrand depending also onxanduis also considered.
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