On extensions of families of operators

Oleg Lihvoinen

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 2, page 227-252
  • ISSN: 0010-2628

Abstract

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The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of r materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions m is less than the dimension n of the reference domain are well-known, but we present several different approaches in this paper to prove that parametrization of the strong closure of feasible states can be done by convexification. Also, a new approach is offered to prove result for the strong closure of cogradients. There are given counterexamples for the case m ≥ n when the parametrization by convexification is not possible. This extends the known result for the case m = n = 2 .

How to cite

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Lihvoinen, Oleg. "On extensions of families of operators." Commentationes Mathematicae Universitatis Carolinae 64.2 (2023): 227-252. <http://eudml.org/doc/299164>.

@article{Lihvoinen2023,
abstract = {The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of the reference domain are well-known, but we present several different approaches in this paper to prove that parametrization of the strong closure of feasible states can be done by convexification. Also, a new approach is offered to prove result for the strong closure of cogradients. There are given counterexamples for the case $m\ge n$ when the parametrization by convexification is not possible. This extends the known result for the case $m=n=2$.},
author = {Lihvoinen, Oleg},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {strong closure; feasible state; operator; elliptic system},
language = {eng},
number = {2},
pages = {227-252},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On extensions of families of operators},
url = {http://eudml.org/doc/299164},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Lihvoinen, Oleg
TI - On extensions of families of operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 2
SP - 227
EP - 252
AB - The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of the reference domain are well-known, but we present several different approaches in this paper to prove that parametrization of the strong closure of feasible states can be done by convexification. Also, a new approach is offered to prove result for the strong closure of cogradients. There are given counterexamples for the case $m\ge n$ when the parametrization by convexification is not possible. This extends the known result for the case $m=n=2$.
LA - eng
KW - strong closure; feasible state; operator; elliptic system
UR - http://eudml.org/doc/299164
ER -

References

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