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25
GaussGreen theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws
 Comm. Pure Appl. Math
"... 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 ..."
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Cited by 19 (4 self)
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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 divergencemeasure field F over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the GaussGreen theorem. To achieve this, we first develop an alternative way to establish the GaussGreen 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 measuretheoretic 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 GaussGreen 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. 1.
Real interpolation of Sobolev spaces
, 2008
"... We prove that W 1 p is an interpolation space between W 1 p1 and W 1 p2 for p> q0 and 1 ≤ p1 < p < p2 ≤ ∞ on some classes of manifolds and general metric spaces, where q0 depends on our hypotheses. ..."
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Cited by 13 (12 self)
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We prove that W 1 p is an interpolation space between W 1 p1 and W 1 p2 for p> q0 and 1 ≤ p1 < p < p2 ≤ ∞ on some classes of manifolds and general metric spaces, where q0 depends on our hypotheses.
Differentiable structures on metric measure spaces: A primer, arXiv 1108.1324
, 2011
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Eigenmodes of a Laplacian on some Laakso Spaces
, 2009
"... We analyze the spectrum of a selfadjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. We will use the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. We also ..."
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Cited by 11 (6 self)
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We analyze the spectrum of a selfadjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. We will use the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. We also extend the method for using the hierarchical cell structure to more general projective limits beyond Laakso spaces. MCS: 34L40 (primary); 34L16; 54B30 1
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
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Cited by 10 (1 self)
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Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
Compactness estimates for the ∂Neumann problem in weighted L 2 spaces on C n , to appear
 in Proccedings of the Conference on Complex Analysis 2008, in honour of Linda Rothschild, arXiv:0903.1783
"... Abstract. In this paper we discuss compactness estimates for the ∂Neumann problem in the setting of weighted L 2spaces on C n. For this purpose we use a version of the Rellich Lemma for weighted Sobolev spaces. 1. Introduction. ..."
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Cited by 9 (8 self)
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Abstract. In this paper we discuss compactness estimates for the ∂Neumann problem in the setting of weighted L 2spaces on C n. For this purpose we use a version of the Rellich Lemma for weighted Sobolev spaces. 1. Introduction.
On the relationship between derivations and measurable differentiable structures
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Dirichlet Forms on Laakso and Barlow Evans Fractals of Arbitrary Dimension
"... In this paper we explore the metricmeasure spaces introduced by Laakso in 2000. Building upon the work of Barlow and Evans we are able to show the existence of a large supply of Dirichlet forms, or alternatively Markov Processes, on these spaces. The construction of Barlow and Evans allows us to ju ..."
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Cited by 2 (2 self)
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In this paper we explore the metricmeasure spaces introduced by Laakso in 2000. Building upon the work of Barlow and Evans we are able to show the existence of a large supply of Dirichlet forms, or alternatively Markov Processes, on these spaces. The construction of Barlow and Evans allows us to justify the use of a quantum graph perspective to identify and describe a Laplacian operator generated by minimal generalized upper gradients on any of Laakso the spaces.
INFINITESIMALLY LIPSCHITZ FUNCTIONS ON METRIC SPACES
, 2009
"... For a metric space X, we study the space D ∞ (X) of bounded functions on X whose infinitesimal Lipschitz constant is uniformly bounded. D ∞ (X) is compared with the space LIP ∞ (X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a B ..."
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Cited by 2 (0 self)
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For a metric space X, we study the space D ∞ (X) of bounded functions on X whose infinitesimal Lipschitz constant is uniformly bounded. D ∞ (X) is compared with the space LIP ∞ (X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a BanachStone theorem in this context. In the case of a metric measure space, we also compare D ∞ (X) with the NewtonianSobolev space N 1, ∞ (X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D ∞ (X) = N 1, ∞ (X).