Journal ArticleDOI
Fine Properties of Functions with Bounded Deformation
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In this article, the fine properties of functions in, the space of functions with bounded deformation, were analyzed, and it was shown that functions are approximately differentiable in almost every point of their domain.Abstract:
The paper is concerned with the fine properties of functions in , the space of functions with bounded deformation. We analyse the set of Lebesgue points and the set where these functions have one-sided approximate limits. Moreover, following the analogy with , we decompose the symmetric distributional derivative into an absolutely continuous part , a jump part , and a Cantor part . The main result of the paper is a structure theorem for functions, showing that these parts of the derivative can be recovered from the corresponding ones of the one-dimensional sections. Moreover, we prove that functions are approximately differentiable in almost every point of their domain.read more
Citations
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Revisiting brittle fracture as an energy minimization problem
TL;DR: In this paper, a variational model of quasistatic crack evolution is proposed, which frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path.
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Regularity Results for Stationary Electro-Rheological Fluids
Emilio Acerbi,Giuseppe Mingione +1 more
TL;DR: In this article, the authors prove regularity results for weak solutions to systems modelling electro-rheological fluids in the stationary case, as proposed in [27, 31].
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A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence
TL;DR: In this paper, a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence is derived, where the scaling of the elastic energy per unit volume is related to the strength of the applied force, where h is the thickness of the plate.
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An approximation result for special functions with bounded deformation
TL;DR: In this article, a special displacement with bounded deformation is approximated with a sequence (un)n⩾1 of piecewise continuous displacements whose jump sets Jun are (relatively) closed, with un and e(un) converging strongly in L2, respectively to u and e.
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A geometrical approach to monotone functions in R n
Giovanni Alberti,Luigi Ambrosio +1 more
TL;DR: In this paper, the fine properties of monotone functions on Ω(n)-convex convex functions were studied. Butler et al. studied the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and the weak Jacobians.
References
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Geometric Measure Theory
TL;DR: In this article, Grassmann algebras of a vectorspace have been studied in the context of the calculus of variations, and a glossary of some standard notations has been provided.
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Journal ArticleDOI
Optimal approximations by piecewise smooth functions and associated variational problems
David Mumford,Jayant Shah +1 more
TL;DR: In this article, the authors introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision, and study their application in computer vision.
Book
Minimal surfaces and functions of bounded variation
TL;DR: In this article, a priori estimation of the gradient of the Bernstein problem is given. But the gradient is not a priorimate of the radius of the singular set, and it is not known whether the gradient can be estimated by direct methods.