arXiv subjects

Piotr Puchała

Publications and source records attributed to Piotr Puchała.

6 recordsLinked to original sources

arXiv2019

Weak convergence of the sequences of homogeneous Young measures associated with a class of oscillating functions

We take under consideration Young measures with densities. The notion of density of a Young measure is introduced and illustrated with examples. It is proved that the density of a Young measure is weakly sequentially closed set. In the case when density of a Young measure is a singleton (up to the set of null measure), it is shown that the strong closedness (in rca(K)) of the set of such measures, associated with Borel functions with values in the compact set K ? Rl, is equivalent with the strong closedness (in L1(K)) of the set of their densities, provided the set K is convex. For an m-oscillating function the notion of a total slope is proposed. It turns out, that if the total slopes of the elements of the sequence of oscillating functions form monotonic sequence, then the sequence of the respective (homogeneous) Young measures is weakly convergent in rca(K). The limit is a homogeneous Young measure with the density being the weak L1 sequential limit of the densities of the underlying Young measures.

math.FA↗

arXiv2017

On general characterization of Young measures associated with Borel functions

We prove that the Young measure associated with a Borel function f is a probability distribution of the random variable f(U), where U has a uniform distribution on the domain of f. As an auxiliary result, the fact that Young measures associated with simple functions are weak* dense in the set of Young measures associated with measurable functions is proved. Finally some examples of specific applications of the main result are presented with comments.

math.FA↗

arXiv2016

A simple characterization of homogeneous Young measures and weak $L^1$ convergence of their densities

We formulate a simple characterization of homogeneous Young measures associated with measurable functions. It is based on the notion of the quasi-Young measure introduced in the previous article published in this Journal. First, homogeneous Young measures associated with the measurable functions are recognized as the constant mappings defined on the domain of the underlying function with values in the space of probability measures on the range of these functions. Then the characterization of homogeneous Young measures via image measures is formulated. Finally, we investigate the connections between weak convergence of the homogeneous Young measures understood as elements of the Banach space of scalar valued measures and the weak* L1 sequential convergence of their densities. A scalar case of the smooth functions and their Young measures being Lebesgue-Stieltjes measures is also analyzed.

math.FA↗

arXiv2014

An elementary method of calculating an explicit form of Young measures in some special cases

We present an elementary method of explicit calculation of Young measures for certain class of functions. This class contains in particular functions of a highly oscillatory nature which appear in optimization problems and homogenization theory. In engineering such situation occurs for instance in nonlinear elasticity (solid-solid phase transition in certain elastic crystals). Young measures associated with oscillating minimizing sequences gather information about their oscillatory nature and therefore about underlying microstructure. The method presented in the paper makes no use of functional analytic tools. There is no need to use generalized version of the Riemann {Lebesgue lemma and to calculate weak* limits of functions. The main tool is the change of variable theorem. The method applies both to sequences of periodic and nonperiodic functions.

math.FA↗

arXiv2013

Nonconvex minimization related to quadratic double-well energy - approximation by convex problems

A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied with taking into account the minimization of the corresponding integral functional. Such integral, as being not sequentially weakly lower semicontinuous, does not admit classical minimizers. To derive the relaxation formula for the infimum, the minimizing sequence consisting of solutions of convex problems appropriately approximating the original nonconvex one is constructed. The weak limit of this sequence together with the weak limit of the sequence of solutions of the corresponding dual problems and the weak limits of the characteristic functions related to the phase energies are involved in the relaxation formula.

math.FA↗

arXiv2005

Continuous version of the Choquet Integral Reperesentation Theorem

The Choquet - Bishop - de Leeuw theorem states that each element of a compact convex subset of a locally convex topological Hausdorff space is a barycenter of a probability measure supported by the set of extreme points of that set. By the Edgar - Mankiewicz result this remains true for nonempty closed bounded and convex set provided it has Radon - Nikodym property. In the paper it is shown, that Choquet - type theorem holds also for "moving" sets: they are values of a certain multifunction. Namely, the existence of a suitable weak* continuous family of probability measures "almost representing" points of such sets is proven. Both compact and noncompact cases are considered. The continuous versions of the Krein - Milman theorem are obtained as corollaries.

math.FA↗