scholarly journals Describing limits of integrable functions as grid functions of nonstandard analysis

2021 ◽  
Vol 2 (4) ◽  
Author(s):  
Emanuele Bottazzi
Keyword(s):  
Functional Analysis ◽  
Nonstandard Analysis ◽  
Young Measure ◽  
Bounded Sequence ◽  
Nonlinear Pdes ◽  
Young Measures ◽  
Grid Function ◽  
Integrable Functions ◽  
Pointwise Limit ◽  
Ill Posed

AbstractIn functional analysis, there are different notions of limit for a bounded sequence of $$L^1$$ L 1 functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $$L^1$$ L 1 functions can be described in terms of its weak-$$\star $$ ⋆ limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson’s nonstandard analysis, we show that for every bounded sequence $$\{z_n\}_{n \in \mathbb {N}}$$ { z n } n ∈ N of $$L^1$$ L 1 functions there exists a function of a hyperfinite domain (i.e. a grid function) that represents both the weak-$$\star $$ ⋆ and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs. We discuss the example of an ill-posed forward–backward parabolic equation.

scholarly journals Projection Methods for Ill-Posed Problems Revisited

2016 ◽  
Vol 16 (2) ◽  
pp. 257-276 ◽  
Author(s):  
Stefan Kindermann
Keyword(s):  
Exact Solution ◽  
Global Convergence ◽  
Least Squares ◽  
Local Convergence ◽  
Sufficient Conditions ◽  
Projection Methods ◽  
Bounded Sequence ◽  
Oblique Projection ◽  
Ill Posed

AbstractWe consider the discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solutions) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent conditions when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as an oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence, we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.

A Young measure approach to a nonlinear membrane model involving the bending moment

2004 ◽  
Vol 134 (5) ◽  
pp. 845-883 ◽  
Author(s):  
Marian Bocea ◽  
Irene Fonseca
Keyword(s):  
Integral Representation ◽  
Young Measure ◽  
Bending Moment ◽  
Complete Characterization ◽  
Membrane Model ◽  
Young Measures ◽  
Probability Measures ◽  
Bending Moments ◽  
Nonlinear Membrane

An integral representation of a relaxed functional arising in the derivation of a nonlinear membrane model accounting for the density of bending moments is obtained in terms of a special class of parametrized probability measures (bending Young measures). A complete characterization of this class is given.

scholarly journals On Simulation of the Young Measures

2018 ◽  
Vol 41 (2) ◽  
pp. 171-184
Author(s):  
Andrzej Z. Grzybowski ◽  
Piotr Puchała
Keyword(s):  
Measure Theory ◽  
Young Measure ◽  
Variational Problems ◽  
Difficult Problem ◽  
Point Of View ◽  
Young Measures ◽  
Regularity Conditions ◽  
Continuous Linear Functional ◽  
Monte Carlo Techniques ◽  
Abstract Notion

"Young measure" is an abstract notion from mathematical measure theory.  Originally, the notion appeared in the context of some variational problems related to the analysis of sequences of “fast” oscillating of functions.  From the formal point of view the Young measure  may be treated as a continuous linear functional defined on the space of Carathéodory integrands satisfying certain regularity conditions. Calculating an explicit form of specific Young measure is a very important task.  However, from a strictly mathematical standpoint  it is a very difficult problem not solved as yet in general. Even more difficult would be the problem of calculating Lebasque’s integrals with respect to such measures. Based on known formal results it can be done only in the most simple cases.  On the other hand in many real-world applications it would be enough to learn only some of the most important probabilistic  characteristics  of the Young distribution or learn only approximate values of the appropriate integrals. In such a case a possible solution is to adopt Monte Carlo techniques. In the presentation we propose three different algorithms designed for simulating random variables distributed according to the Young measures  associated with piecewise functions.  Next with the help of computer simulations we compare their statistical performance via some benchmarking problems. In this study we focus on the accurateness of the distribution of the generated sample.

scholarly journals Asymptotic farthest points and extreme points

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5875-5885
Author(s):  
Nejhad Ardakani ◽  
Mazaheri Tehrani
Keyword(s):  
Functional Analysis ◽  
Approximation Theory ◽  
Normed Space ◽  
Extreme Points ◽  
Bounded Sequence ◽  
Bounded Subset ◽  
Farthest Points ◽  
Farthest Point

Let X be a normed space, G a nonempty bounded subset of X and fxng a bounded sequence in X. In this article, we introduce and discuss the concept of asymptotic farthest points of fxng in G, which is a new definition in abstract approximation theory. Then, by applying the topics of functional analysis, we investigate the relation between this new concept and the concepts of extreme points and convexity. In particular, one of the main purposes of this paper is to study conditions under which the existence (uniqueness) of asymptotic farthest point of fxng in G is equivalent to the existence (uniqueness) of asymptotic farthest point of fxng in ext(G) or co(G).

scholarly journals Theory of hypernumbers and extrafunctions: Functional spaces and differentiation

2002 ◽  
Vol 7 (3) ◽  
pp. 201-212 ◽  
Author(s):  
Mark Burgin
Keyword(s):  
Functional Analysis ◽  
Nonstandard Analysis ◽  
Computational Physics ◽  
New Technique ◽  
Complex Numbers ◽  
General Version ◽  
Novel Approach ◽  
External Objects ◽  
Convergence And Divergence

The theory of hypernumbers and extrafunctions is a novel approach in functional analysis aimed at problems of mathematical and computational physics. The new technique allows operations with divergent integrals and series and makes it possible to distinct different kinds of convergence and divergence. Although, it resembles nonstandard analysis, there are several distinctions between these theories. For example, while nonstandard analysis changes spaces of real and complex numbers by injecting into them infinitely small numbers and other nonstandard entities, the theory of extrafunctions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big and oscillating numbers as external objects. In this paper, we consider a simplified version of hypernumbers, but a more general version of extrafunctions and their extraderivatives in comparison with previous works.

scholarly journals Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem

2019 ◽  
Vol 25 ◽  
pp. 26
Author(s):  
Francesco Della Porta
Keyword(s):  
Model Problem ◽  
Young Measure ◽  
Young Measures ◽  
Selection Mechanism ◽  
Minimizing Sequences ◽  
Dimensional Model ◽  
One Dimensional ◽  
Energy Functionals ◽  
Second Derivatives ◽  
Gradient Young Measure

Energy functionals describing phase transitions in crystalline solids are often non-quasiconvex and minimizers might therefore not exist. On the other hand, there might be infinitely many gradient Young measures, modelling microstructures, generated by minimizing sequences, and it is an open problem how to select the physical ones. In this work we consider the problem of selecting minimizing sequences for a one-dimensional three-well problem ε. We introduce a regularization εε of ε with an ε-small penalization of the second derivatives, and we obtain as ε ↓ 0 its Γ-limit and, under some further assumptions, the Γ-limit of a suitably rescaled version of εε. The latter selects a unique minimizing gradient Young measure of the former, which is supported just in two wells and not in three. We then show that some assumptions are necessary to derive the Γ-limit of the rescaled functional, but not to prove that minimizers of εε generate, as ε ↓ 0, Young measures supported just in two wells and not in three.

scholarly journals Interchanging a limit and an integral: necessary and sufficient conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Takashi Kamihigashi
Keyword(s):  
Measure Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Integrable Functions ◽  
Finite Measure Space ◽  
Pointwise Limit ◽  
Necessary And Sufficient

AbstractLet $\{f_{n}\}_{n \in \mathbb {N}}$ { f n } n ∈ N be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ ( Ω , F , μ ) . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ lim n ↑ ∞ f n exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$ lim n ↑ ∞ ∫ f n d μ = ∫ lim n ↑ ∞ f n d μ .

scholarly journals Weak solutions for generalized p-Laplacian systems via Young measures

2018 ◽  
Vol 4 (2) ◽  
pp. 77-84 ◽  
Author(s):  
Elhoussine Azroul ◽  
Farah Balaadich
Keyword(s):  
Weak Solutions ◽  
Existence Result ◽  
Young Measure ◽  
Elliptic Systems ◽  
Young Measures ◽  
Existence Of Weak Solutions ◽  
Degenerate Form

AbstractWe prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.

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