<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msup></mml:math>-variational problems associated to measurable Finsler structures

Guo, Chang‐Yu; Xiang, Chang‐Lin; Yang, Dachun

doi:10.1016/j.na.2015.10.025

Cited by 8 publications

(13 citation statements)

References 31 publications

(58 reference statements)

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“…If H(x, p) is given b A(x)p, p with A being as in Remark 1.5 (ii) above, existence of absolute minimizers is given by [23] with the aid of [22]. In a similar way, with the aid of [22], Guo et al [18] also obtained the the existence of absolute minimizers if H(x, p) is a measurable function in Ω × R n , and satisfies that 1 C < H(x, p) < C for all x ∈ Ω and p ∈ S n−1 , where C ≥ 1 is a constant, and that H(x, ηp) = |η|H(x, p) for all x ∈ Ω, p ∈ R n and η ∈ R.…”

Section: Introductionmentioning

confidence: 99%

A Rademacher type theorem for Hamiltonians $H(x,p)$ and application to absolute minimizers

Liu¹,

Zhou²

2022

Preprint

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We establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory spaces. In particular, H(x, p) is assumed to be only measurable in the variable x, and to be quasiconvex and lower-semicontinuous in the variable p; the lower-simicontinuity in the variable p is sharp. Moreover, applying such Rademacher type theorem we build up an existence result of absolute minimizers for corresponding L ∞ -functional. These improve or extend several known results in the literature.

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Section: Introductionmentioning

confidence: 99%

A Rademacher type theorem for Hamiltonians $H(x,p)$ and application to absolute minimizers

Liu¹,

Zhou²

2022

Preprint

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“…Since then, various works have been devoted to the analysis of Δ ∞ for more general structures, and an account can be found in [21,9]. Especially, on domains of ℝ equipped with a Finsler norm, the AMLE problem and the associated ∞-Laplace operator have been studied in [55,29,41,42]. One of the starting points of the present investigation is the following Liouville theorem for ∞-subharmonic functions on ℝ (cf.…”

Section: Introductionmentioning

confidence: 99%

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

Araújo,

Mari,

Pessoa

2020

Preprint

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In this paper, we study some potential theoretic aspects of the eikonal and infinity Laplace operator on a Finsler manifold . Our main result shows that the forward completeness of can be detected in terms of Liouville properties and maximum principles at infinity for subsolutions of suitable inequalities, including Δ ∞ ≥ ( ). Also, an ∞-capacity criterion and a viscosity version of Ekeland principle are proved to be equivalent to the forward completeness of . Part of the proof hinges on a new boundary-to-interior Lipschitz estimate for solutions of Δ ∞ = ( ) on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of .

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“…Our approach is more geometric and was influented a lot by the recent studies in Finsler geometry [6,7,2,4]. Some of the ideas from this paper were successfully used in our companion paper [9] on certain L ∞ -variational problems associated to measurable Finsler structures. It is known (e.g.…”

Section: Introductionmentioning

confidence: 99%