Intrinsic geometry and analysis of Finsler structures

Guo, Chang‐Yu

doi:10.1007/s10231-017-0634-7

Cited by 4 publications

(5 citation statements)

References 10 publications

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“…We point out that inhomogeneous versions of (1.1) appeared already in [26], and more lately in [28,31]. Moreover, it is useful to observe that functionals of the type (1.1) share key features with two different classes of functionals that have been studied intensively in the literature, namely double-integral functionals mentioned already at the beginning, i.e., L p ( ; R m ) u → W (u(x), u(y)) dx dy with p ∈ [1, ∞), and supremal functionals (or L ∞ -functionals), i.e.,…”

Section: Introductionmentioning

confidence: 71%

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Kreisbeck

Zappale

2020

Calc. Var.

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We study variational problems involving nonlocal supremal functionalswhere ⊂ R n is a bounded, open set and W : R m × R m → R is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for L ∞ -weak * lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. The analogous statement in the related context of double-integral functionals was recently shown to be false. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak * closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.

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Section: Introductionmentioning

confidence: 71%

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Kreisbeck

Zappale

2020

Calc. Var.

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“…The proof is similar to that of[23, Proposition 3.1 ]. The inequality δ F (x, y) ≤ d * c (x, y) follows directly from definition.…”

mentioning

confidence: 61%

“…In the appendix, we give an alternative proof of Lemma 2.5 based on an approximation argument similar to the proof of [23,Proposition 3.1]. The proof is based on a personal communication with Professor Andrea Davini.…”

Section: Proof Of Existencementioning

confidence: 99%

“…For the existence result, our proof relies on the (crucial) Lemma 4.2 and Proposition 3.1, which allows us to describe the absolute minimizer via the pointwise Lipschitz constant. In this step, we also borrow some ideas from the recent related work [23], which allow us to relate the geometric and the analytic aspects of admissible Finsler structures.…”

Section: Introductionmentioning

confidence: 99%

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L∞-variational problems associated to measurable Finsler structures

2016

Self Cite

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“…We point out that inhomogeneous versions of (1.1) appeared already in [26], and more lately in [28,31]. Moreover, it is useful to observe that functionals of the type (1.1) share key features with two different classes of functionals that have been studied intensively in the literature, namely double-integral functionals mentioned already at the beginning, i.e., L p (Ω; R m ) ∋ u → Ω Ω W (u(x), u(y)) dx dy with p ∈ [1, ∞), and supremal functionals (or L ∞ -functionals), i.e.,…”

Section: Introductionmentioning

confidence: 71%

Lower semicontinuity and relaxation of nonlocal $L^\infty$-functionals

Kreisbeck,

Zappale

2019

Preprint

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We study variational problems involving nonlocal supremal functionalswhere Ω ⊂ R n is a bounded, open set and W : R m × R m → R is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for L ∞weak * lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. The analogous statement in the related context of double-integral functionals was recently shown to be false. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak * closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.

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Intrinsic geometry and analysis of Finsler structures

Abstract: In this short note, we prove that if F is a weak upper semicontinuous admissible Finsler structure on a domain in R n , n ≥ 2, then the intrinsic distance and differential structures coincide.

Cited by 4 publications

References 10 publications

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

L∞-variational problems associated to measurable Finsler structures

Lower semicontinuity and relaxation of nonlocal $L^\infty$-functionals

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