Results 1 
9 of
9
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 ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
INFINITESIMALLY LIPSCHITZ FUNCTIONS ON METRIC SPACES
, 901
"... Abstract. 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 ob ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. 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). 1.
RADEMACHER’S THEOREM IN R n
"... The purpose of these notes is to prove the classical Rademacher’s Theorem. There are versions that are far more general, notably Cheeger obtained recently a Rademacher type theorem only assuming the space has a Vitali Covering Theorem, which incidentally gives a new proof of the classical ..."
Abstract
 Add to MetaCart
The purpose of these notes is to prove the classical Rademacher’s Theorem. There are versions that are far more general, notably Cheeger obtained recently a Rademacher type theorem only assuming the space has a Vitali Covering Theorem, which incidentally gives a new proof of the classical
Some aspects of calculus on nonsmooth sets
, 709
"... Let E be a closed set in R n, and suppose that there is a k ≥ 1 such that every x, y ∈ E can be connected by a rectifiable path in E with length ≤ k x−y. This condition is satisfied by chordarc curves, Lipschitz manifolds of any dimension, and fractals like Sierpinski gaskets and carpets. Note th ..."
Abstract
 Add to MetaCart
Let E be a closed set in R n, and suppose that there is a k ≥ 1 such that every x, y ∈ E can be connected by a rectifiable path in E with length ≤ k x−y. This condition is satisfied by chordarc curves, Lipschitz manifolds of any dimension, and fractals like Sierpinski gaskets and carpets. Note that lengthminimizing paths in E are chordarc curves with constant k. A basic feature of this condition is that one can integrate local Lipschitz conditions on E to get global conditions. For instance, if f: E → R is locally Lipschitz of order 1 with constant C ≥ 0, then f is globally Lipschitz on E with constant k C. Let A(x) be a continuous function on E with values in linear mappings from Rn to R. Equivalently, one can use the standard inner product on Rn to represent A(x) by a continuous mapping from E into Rn. Also let f be a locally Lipschitz realvalued function on E. Suppose that A includes the directional derivatives of f almost everywhere on any rectifiable curve in E, in the sense that the derivative of f(p(t)) is equal to A(p(t)) applied to the derivative of p(t) for almost every t when p(t) is a
Analysis on disconnected sets
, 709
"... Very often in analysis, one focuses on connected spaces. This is certainly not always the case, and in particular there are many interesting matters related to Cantor sets. Here we are more concerned with a type of complementary situation. As a basic scenario, suppose that U is an open set in R n an ..."
Abstract
 Add to MetaCart
Very often in analysis, one focuses on connected spaces. This is certainly not always the case, and in particular there are many interesting matters related to Cantor sets. Here we are more concerned with a type of complementary situation. As a basic scenario, suppose that U is an open set in R n and that E is a closed set contained in the boundary of U such that for every x∈U and r> 0 there is a connected component of U contained in the ball with center x and radius r. For instance, E might be the boundary of U. As a uniform version of the condition, one might ask that there be a constant C> 0 such that the aforementioned connected component of U contains a ball of radius C −1 r. The connected components of U might be quite regular, even if there are infinitely many of them. As a basic example, E could be a Cantor set in the real line, and U could be the complement of E or the union of the bounded complementary components of E. This does not work in higher dimensions, where the complement of a Cantor set is connected. Sierpinski gaskets and carpets in the plane are
Dirichlet Forms on Laakso and Some BarlowEvans Fractals of Arbitrary Dimension
, 2009
"... In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincaré inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upp ..."
Abstract
 Add to MetaCart
In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincaré inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger’s work these upper gradients can be used to define a Sobolev space. We show that this leads to a Dirichlet form. The second construction was introduced by Barlow and Evans in 2004 as a way of producing exotic spaces along with Markov processes from simpler spaces and processes. We show that for the correct base process in the Barlow Evans construction that this Markov process corresponds to the Dirichlet form derived from the minimal generalized upper gradients. MSC Codes: 31C25 (Primary) 60J45, 28A80, 46A13 1