Some remarks on \(\Gamma\)-convergence and least squares method.

*(English)*Zbl 0747.49008
Composite media and homogenization theory, Proc. Int. Cent. Theor. Phys. Workshop, Trieste/Italy 1990, Prog. Nonlinear Differ. Equ. Appl. 5, 135-142 (1991).

[For the entire collection see Zbl 0722.00038.]

Ideas and conjectures are given on how to use the methods of \(\Gamma\)- convergence of functionals in the study of differential equations (Originally, the theory of \(\Gamma\)-convergence was especially developed by the author and others to handle problems related to optimization and similar topics). Reformulating the given differential equations by least squares terms with some parameter investigations of their weak solutions can be taken by using \(\Gamma\)-limits of such terms with respect to the parameter considering weak topologies. In a few illustrative examples the main ideas and conjectures are explained.

The author’s intention of giving new stimulations to specialists for further researches in this field of weak solutions for differential equations can be regarded as very successful.

Ideas and conjectures are given on how to use the methods of \(\Gamma\)- convergence of functionals in the study of differential equations (Originally, the theory of \(\Gamma\)-convergence was especially developed by the author and others to handle problems related to optimization and similar topics). Reformulating the given differential equations by least squares terms with some parameter investigations of their weak solutions can be taken by using \(\Gamma\)-limits of such terms with respect to the parameter considering weak topologies. In a few illustrative examples the main ideas and conjectures are explained.

The author’s intention of giving new stimulations to specialists for further researches in this field of weak solutions for differential equations can be regarded as very successful.

Reviewer: A.Hoffmann (Ilmenau)

##### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

35D99 | Generalized solutions to partial differential equations |

35A15 | Variational methods applied to PDEs |

54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |

49J45 | Methods involving semicontinuity and convergence; relaxation |