#### DMCA

## with wide range of anisotropy (2014)

### Citations

851 |
Interpolation Theory, Function Spaces, Differential Operators
- Triebel
- 1978
(Show Context)
Citation Context ...ave ∫ Ω ∫ f τs,hg dx = − g τs,−hf dx. Ω τs,h(fg)(x) = f(x + hes)τs,hg(x) + g(x)τs,hf(x). We also recall fundamental embedding properties for fractional Sobolev spaces (for the proof see, for example, =-=[19]-=-). 6Lemma 1. If w : R n → R N , w ∈ L p (BR) with 1 < p < n and for some ρ ∈ (0, R), β ∈ (0, 1], M > 0, n∑ ∫ s=1 for every h with |h| < R−ρ k ∈ (0, β) and ‖w‖ L with c ≡ c(n, N, R, ρ, β, k). Bρ |τs,h... |

705 |
Convex Analysis and Variational Problems
- Ekeland, Temam
- 1976
(Show Context)
Citation Context ...gh |F ′ (ξ)| will always denote the euclidean norm of the matrix F ′ (ξ). We shall require some further elementary notions from convex analysis, all of which are discussed in the scalar case N = 1 in =-=[7]-=-. However, as we shall briefly recall below, the relevant parts easily extend to the vectorial case N > 1 too. Given F : R N×n → R, its polar (or Fenchel conjugate) integrand is defined by F ∗ (ζ) := ... |

377 |
Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic
- Giaquinta
- 1983
(Show Context)
Citation Context ...stigated in the literature. From very early on it has been clear that little regularity can be expected if the coercitivity and growth exponents, denoted p and q, respectively, are too far apart (see =-=[14, 10]-=-). On the other hand, many regularity results are available if the ratio q/p is bounded above by a suitable constant depending on the dimension n, and converging to 1 when n tends to infinity (incl. [... |

52 |
Direct methods in the calculus of variations, World Scientific Publishing
- Giusti
- 2003
(Show Context)
Citation Context ...rence quotient is defined for h ∈ R \ {0} as ∆s,hw(x) = τs,hw(x) . h The following proposition describes some elementary properties of the finite difference operator and can be found, for example, in =-=[11]-=-. Proposition 1. Let f and g be two functions such that f, g ∈ W 1,p (Ω), with p ≥ 1, and let us consider the set Then (d1) τs,hf ∈ W 1,p (Ω) and Ω |h| := {x ∈ Ω : dist(x, ∂Ω) > |h|} . Di(τs,hf) = τs,... |

51 | Differentiable Functions, Graduate Texts in Mathematics, Vol. 120 - Ziemer, Weakly - 1989 |

40 | Regularity of minima: An invitation to the dark side of the calculus of variations - Mingione - 2006 |

36 | The singular set of minima of integral functionals - Kristensen, Mingione - 2006 |

26 |
Regularity results for minimizers of irregular integrals with (p, q)-growth
- Esposito, Leonetti, et al.
(Show Context)
Citation Context ...]). On the other hand, many regularity results are available if the ratio q/p is bounded above by a suitable constant depending on the dimension n, and converging to 1 when n tends to infinity (incl. =-=[1, 2, 3, 8, 9, 16, 18]-=-). Note that here we do not impose any growth condition from above and we only suppose a very weak condition from below. Nevertheless, having at our disposal Theorem 1, we are able to establish the fo... |

24 |
Partial regularity under anisotropic (p, q) growth conditions
- Acerbi, Fusco
- 1994
(Show Context)
Citation Context ...]). On the other hand, many regularity results are available if the ratio q/p is bounded above by a suitable constant depending on the dimension n, and converging to 1 when n tends to infinity (incl. =-=[1, 2, 3, 8, 9, 16, 18]-=-). Note that here we do not impose any growth condition from above and we only suppose a very weak condition from below. Nevertheless, having at our disposal Theorem 1, we are able to establish the fo... |

23 | Nonlinear elliptic systems with general growth
- Marcellini, Papi
(Show Context)
Citation Context ...]). On the other hand, many regularity results are available if the ratio q/p is bounded above by a suitable constant depending on the dimension n, and converging to 1 when n tends to infinity (incl. =-=[1, 2, 3, 8, 9, 16, 18]-=-). Note that here we do not impose any growth condition from above and we only suppose a very weak condition from below. Nevertheless, having at our disposal Theorem 1, we are able to establish the fo... |

16 |
Higher integrability for minimizers of integral functionals with (p, q)-growth
- Esposito, Leonetti, et al.
- 1999
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Citation Context |

8 | Boundary regularity in variational problems - Kristensen, Mingione |

7 |
A regularity result for a class of anisotropic systems
- Napoli, Siepe
- 1996
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Citation Context |

2 |
Regularity results via duality for minimizers of degenerate functionals
- Carozza, Moscariello, et al.
(Show Context)
Citation Context ... we can associate a row-wise solenoidal matrix field denoted by σk. For the pair (Duk, σk) we shall establish suitable estimates, that are preserved in passing to the limit (compare with [4] and also =-=[5]-=-, [6] for a related approach in the case of degenerate convex problems of standard growth). The study of the regularity properties of minimizers of integral functionals at (1.1) often comes through th... |

2 |
di Napoli. Regularity for minimizers of degenerate elliptic functionals
- Carozza, Passarelli
(Show Context)
Citation Context ...an associate a row-wise solenoidal matrix field denoted by σk. For the pair (Duk, σk) we shall establish suitable estimates, that are preserved in passing to the limit (compare with [4] and also [5], =-=[6]-=- for a related approach in the case of degenerate convex problems of standard growth). The study of the regularity properties of minimizers of integral functionals at (1.1) often comes through their e... |

1 |
Higher differentiability of minimizers of convex variational integrals
- Carozza, Kristensen, et al.
(Show Context)
Citation Context |

1 |
di Napoli. Regularity of minimizers of autonomous convex variational integrals
- Carozza, Kristensen, et al.
(Show Context)
Citation Context ... provided the growth exponents are related by Ω p ≤ q < p + 1. (1.2) In fact, under the assumption (1.2) the field F ′(Du) is integrable, actually p/(q − 1)–integrable, and is row–wise solenoidal. In =-=[4]-=- we were able to remove the growth assumption from above. In fact, we showed that if F is a convex integrand, the minimality implies the extremality of a W 1,p g map, just imposing a p–convexity condi... |

1 |
example de solution discontinue d’ un probéme variationel dans le cas scalaire. Preprint Ist. U.Dini
- Un
- 1987
(Show Context)
Citation Context ...stigated in the literature. From very early on it has been clear that little regularity can be expected if the coercitivity and growth exponents, denoted p and q, respectively, are too far apart (see =-=[14, 10]-=-). On the other hand, many regularity results are available if the ratio q/p is bounded above by a suitable constant depending on the dimension n, and converging to 1 when n tends to infinity (incl. [... |