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3 Journal of Applied Mathematics and Computational Mechanics

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Certain convergence results for homogeneous singular young measures

Puchała Piotr

Journal of Applied Mathematics and Computational Mechanics

2023

Vol. 22, nr 4

44--52

We consider purely singular homogeneous Young measures associated with elements of sequences of piecewise constant functions and with limits of such sequences. We first consider a case when the limit of a such sequence is piecewise constant. The next point involves the sequences of bounded oscillating functions, divergent in the strong topology in L ∞ , but weakly∗ convergent to a homogeneous Young measure. We also present an example of a fast oscillating sequence, illustrating the result. In the presented results, generalizing to some extent known examples, we try to avoid advanced methods of functional analysis that are usually used when solving problems of this type.

On a certain embedding in the space of measures

Puchała Piotr

Journal of Applied Mathematics and Computational Mechanics

2021

Vol. 20, nr 2

53--63

We take under consideration Young measures - objects that can be interpreted as generalized solutions of a class of certain nonconvex optimization problems arising among others in nonlinear elasticity or micromagnetics. They can be looked at from several points of view. We look at Young measures as at a class of weak* measurable, measure-valued mappings and consider the basic existence theorem for them. On the basis of this theorem, an imbedding of the set of bounded Borel functions into the set of Young measures is defined. Using the weak* denseness of the set of Young measures associated with simple functions in the set of Young measures, it is shown that this imbedding assigns the Young measure associated with any bounded Borel function.

A few remarks on an embedding into the set of measures

Puchała Piotr

Journal of Applied Mathematics and Computational Mechanics

2021

Vol. 20, nr 4

77--85

We continue considerations concerning Young measures associated with bounded measurable functions from a recent article. We look at them as at the weak* measurable, measure-valued mappings. We show examples explaining that we cannot regard a Young measure (i.e. a weak* -measurable mapping) δu(x) as an explicit form of a Young measure associated with a function u. We also consider convergence of the sequences of Young measures.