Abstract
We discuss some results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. The first case concerns a larger class of integrands and requires the positivity almost everywhere of the difference between the upper and lower constraints. In the second case, this requirement is absent. Moreover, in the latter case, the exhaustion condition of an n-dimensional domain by a sequence of n-dimensional domains plays an important role. We give a series of results involving this condition. In particular, using the exhaustion condition, we prove a certain convergence of sets of functions defined by bilateral (generally irregular) constraints in variable domains.
Highlights
This paper is mainly based on the talk given by the author at the International S.B
We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function
Let condition (∗1) of Theorem 1 be satisfied, and assume that there exists a sequence of linear continuous operators ls : W 1,p(Ωs) → W 1,p(Ω) such that the sequence of norms ls is bounded and, for every s ∈ N and for every v ∈ W 1,p(Ωs), we have qs(lsv) = v a.e. in Ωs
Summary
This paper is mainly based on the talk given by the author at the International S.B. Stechkin Summer Workshop-Conference on Function Theory, Miass, Russia, August 1–10, 2017. If the space W is reflexive, there exist an increasing sequence {sj} ⊂ N and an element u ∈ W such that lsj usj → u weakly in W This is the first step in the study of the convergence of the sequence of minimizers us ∈ Ws of the functionals Is. The described idea with the operators ls is realized in the justification of the results stated below for functionals defined on the Sobolev spaces W 1,p(Ωs), where {Ωs} is a sequence of domains contained in a bounded domain Ω of Rn. Essentially, the mentioned idea goes back to [8]. We consider the notion of H-convergence of sequences of sets Us ⊂ W 1,p(Ωs) to a set U ⊂ W 1,p(Ω) and show the importance of the exhaustion condition for the H-convergence of sets of functions defined by irregular bilateral constraints
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