Sharp regularity for functionals with (p,q) growth

Esposito, Luca; Leonetti, Francesco; Mingione, Giuseppe

doi:10.1016/s0022-0396(04)00208-6

Cited by 99 publications

(171 citation statements)

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“…Regularity results. In the recent papers [15,16] the last two named authors have investigated the regularity properties of local minimisers of the functional P p,q in (1.4) under sharp regularity assumptions on the modulating coefficient a(•) (see [8,28] for previous contributions on this functional). The main outcome is that there is a sharp interaction between the regularity properties of a(•) and the gap q/p.…”

Section: Introduction Results and The Functional Settingmentioning

confidence: 99%

Non-autonomous functionals, borderline cases and related function classes

Baroni

Colombo

Mingione

2016

St. Petersburg Math. J.

228

116

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We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p, q)-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima.

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Section: Introduction Results and The Functional Settingmentioning

confidence: 99%

Non-autonomous functionals, borderline cases and related function classes

Baroni

Colombo

Mingione

2016

St. Petersburg Math. J.

228

116

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“…16) and then the sequence Dv " is bounded in L n Q p n 2 .B R ; R N n /, for all Q p < p, and a standard diagonal argument gives a subsequence v j D .v " / j weakly converging to a function v whose gradient Dv satisfies Dv 2 L n Q p n 2 .B R ; R N n /; 8 Q p < p: 16) and then the sequence Dv " is bounded in L n Q p n 2 .B R ; R N n /, for all Q p < p, and a standard diagonal argument gives a subsequence v j D .v " / j weakly converging to a function v whose gradient Dv satisfies Dv 2 L n Q p n 2 .B R ; R N n /; 8 Q p < p:…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Higher differentiability of minimizers of variational integrals with Sobolev coefficients

Napoli

2014

Advances in Calculus of Variations

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In this paper we consider integral functionals of the formwith convex integrand satisfying p growth conditions with respect to the gradient variable. As a novel feature, the dependence of the integrand on the x-variable is allowed to be through a Sobolev function. We prove local higher differentiability results for local minimizers of the functional F, establishing uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand. Furthermore, we prove a dimension free higher integrability result for the gradient of local minimizers, by the use of a weighted version of the Gagliardo-Nirenberg interpolation inequality.

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“…In the vectorial case and in the case of scalar functionals depending also on the state variable there are examples that show that one can not expect that the minimum is locally Lipschitz ( [5], [16]). This is in fact the crucial property to prove the validity of Euler equation (at least under very weak regularity assumptions on F ) and hence to apply the De Giorgi-Moser-Nash Theorem.…”

Section: Introductionmentioning

confidence: 99%