Nonsmooth calculus

Heinonen, Juha

doi:10.1090/s0273-0979-07-01140-8

Cited by 105 publications

(89 citation statements)

References 125 publications

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“…The term "almost all" is expressed in terms of a condition that resembles "extremal length", a concept used in complex analysis and potential theory (cf. [Zie67,Zie69,Zie70,He07]). One way of summarizing our work in this paper is to say that we wish to extend Fuglede's result so that (2.1) holds for every set E of finite perimeter.…”

Section: Introductionmentioning

confidence: 99%

Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

Chen

Torres

Ziemer

2008

Comm Pure Appl Math

123

167

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Abstract. We analyze a class of weakly differentiable vector fields F : R N → R N with the property that F ∈ L ∞ and div F is a Radon measure. These fields are called bounded divergencemeasure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we first develop an alternative way to establish the Gauss-Green theorem for any smooth bounded set with F ∈ L ∞ . Then we establish a fundamental approximation theorem which states that, given a Radon measure µ that is absolutely continuous with respect to H N−1 on R N , any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure µ . We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter, E, as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for F holds on E.With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an (N − 1)-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measurevalued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.

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Section: Introductionmentioning

confidence: 99%

Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

Chen

Torres

Ziemer

2008

Comm Pure Appl Math

123

167

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“…Independently of such physical considerations, there has been a recent surge of activity in Geometry and Analysis purporting to better understand first order calculus properties of non-smooth spaces [21]. An influential work in this direction has been [22], where conditions were given for the existence of a differentiable structure on metric measure spaces, based on Lipschitz maps, which also have the doubling property and admit a Poincaré inequality.…”

Section: Reflexive and Super-reflexive Spacesmentioning

confidence: 99%

Convexity and the Euclidean Metric of Space-Time

Kalogeropoulos

2017

Universe

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Abstract:We address the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

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“…The search for regularity conditions on a metric measure space allowing to generalize results and concepts of first order calculus, for example the notions of derivative and gradient, has been a topic of intensive research. We refer the reader to the survey [Hei07] for more details. A fundamental result about the geometry of Lipschitz functions on Euclidean spaces is the Rademacher Differentiation Theorem which asserts that a Lipschitz function is differentiable a.e.…”

Section: Introductionmentioning

confidence: 99%