NON-LOCAL VARIATIONAL LIMITS OF DISCRETE SYSTEMS

2000 ◽  
Vol 02 (02) ◽  
pp. 285-297 ◽  
Author(s):  
ANDREA BRAIDES
Keyword(s):  
Bounded Variation ◽  
Special Functions ◽  
Discrete Systems ◽  
Functions Of Bounded Variation ◽  
Step Size ◽  
One Dimensional ◽  
Continuum Energy ◽  
Non Local ◽  
Local Term

We show that the limit of a class of discrete energies defined on one-dimensional lattices of step size ε when ε→0 define a continuum energy with a local and a non-local term with domain a subspace of the special functions of bounded variation.

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750048 ◽  
Author(s):  
Y. S. LIANG
Keyword(s):  
Fractal Dimension ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Nondifferentiable Functions ◽  
Differentiable Functions ◽  
Closed Intervals ◽  
Bounded Variation Functions

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

scholarly journals Which special functions of bounded deformation have bounded variation?

2017 ◽  
Vol 148 (1) ◽  
pp. 33-50 ◽  
Author(s):  
Sergio Conti ◽  
Matteo Focardi ◽  
Flaviana Iurlano
Keyword(s):  
Bounded Variation ◽  
Special Functions ◽  
Functions Of Bounded Variation ◽  
Finite Area ◽  
Piecewise Affine ◽  
Positive Side ◽  
Almost Everywhere ◽  
Functions Of Bounded Deformation ◽  
Additional Regularity ◽  
Jump Set

Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.

Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials

2008 ◽  
Vol 138 (5) ◽  
pp. 1019-1041 ◽  
Author(s):  
Alessandro Giacomini ◽  
Marcello Ponsiglione
Keyword(s):  
Fracture Mechanics ◽  
Weak Convergence ◽  
Bounded Variation ◽  
Special Functions ◽  
Brittle Materials ◽  
Functions Of Bounded Variation ◽  
Variational Models ◽  
Stability Results

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

THE CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR FRACTIONAL INTEGRAL

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850063 ◽  
Author(s):  
XING LIU ◽  
JUN WANG ◽  
HE LIN LI
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional

This paper mainly discusses the continuous functions whose fractal dimension is 1 on [Formula: see text]. First, we classify continuous functions into unbounded variation and bounded variation. Then we prove that the fractal dimension of both continuous functions of bounded variation and their fractional integral is 1. As for continuous functions of unbounded variation, we solve several special types. Finally, we give the example of one-dimensional continuous function of unbounded variation.

scholarly journals Functions of Bounded Variation on Complete and Connected One-Dimensional Metric Spaces

2020 ◽  
Author(s):  
Panu Lahti ◽  
Xiaodan Zhou
Keyword(s):  
Bounded Variation ◽  
Hausdorff Measure ◽  
Metric Spaces ◽  
General Setting ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Underlying Space ◽  
Finite Perimeter ◽  
Bv Functions ◽  
Definition Of

Abstract In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda [18]. Furthermore, we study the necessity of conditions on the underlying space in Federer’s characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

scholarly journals The Γ-limit for singularly perturbed functionals of Perona–Malik type in arbitrary dimension

2014 ◽  
Vol 24 (06) ◽  
pp. 1091-1113 ◽  
Author(s):  
Giovanni Bellettini ◽  
Antonin Chambolle ◽  
Michael Goldman
Keyword(s):  
Special Functions ◽  
Arbitrary Dimension ◽  
Convergence Result ◽  
Singularly Perturbed ◽  
Functions Of Bounded Variation ◽  
Second Derivative ◽  
One Dimensional ◽  
Density Result ◽  
Arbitrary Dimensions ◽  
Γ Convergence

In this paper, we generalize to arbitrary dimensions a one-dimensional equicoerciveness and Γ-convergence result for a second derivative perturbation of Perona–Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a "cohesive" energy, that is, whose cost depends on the actual opening of the discontinuity.

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