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**Variational convergence for functions and operators.**
*(English)*
Zbl 0561.49012

Applicable Mathematics Series. Boston - London - Melbourne: Pitman Advanced Publishing Program. XIX, 423 p. £19.95 (1984).

It is commonly agreed that the epigraphical resp. hypographical approach is the most appropriate one in minimization resp. maximization problems involving extended-real-valued functions. Consequently, when studying limits of sequences of such optimization problems it is quite natural to adopt this approach which yields the so-called epi- resp. hypo- convergence of functions. These concepts are particular cases of ”variational convergences” which have been introduced and studied by the optimizing community during the last two decades. It is undoubtedly the merit of the author to clarify these convergence concepts and to illustrate their wide range of applicability in practical problems.

The book consists of three chapters. Chapter 1 introduces epi-convergence in a general topological setting and states its variational properties. Applications cover such difficult-to-handle limit processes as homogenization problems and singularly perturbed optimal control problems. Further, epi-convergence is geometrically interpreted as set- convergence of epigraphs and its relation to the concept of \(\Gamma\)- convergence, developed by De Giorgi’s Italian school, is pointed out.

In Chapter 2 topological properties of epi-convergence such as lower semicontinuity of epi-limits are studied, compactness results are given and epi-limits are compared for different topologies. In particular, it is shown that the epi-limit of a sequence of functions can be characterized as pointwise limit of their Moreau-Yosida approximates. Applications are taken from variational inequalities with varying obstacles, optimal control design and stochastic homogenization.

In Chapter 3 epi-convergence of convex functions on reflexive Banach spaces is studied with special emphasis on its relation to the Mosco- convergence and to the graph-convergence of subdifferential operators. It is shown that epi-convergence of a sequence of convex functions with respect to the weak topology is equivalent to the epi-convergence of their Yound-Fenchel transformations in the strong topology. From that result it is easily revealed that Mosco-convergence is equivalent to epi- convergence for both strong and weak topologies. The results are applied to a singularly perturbed optimal control problem and to the limit analysis for variational inequalities with obstacle constraints. Concerning subdifferential operators it is proved by means of Moreau- Yosida approximation that Mosco-convergence of a sequence of closed convex proper functions is equivalent to pointwise convergence of the resolvents of the corresponding subdifferential operators. A similar result in terms of epi-convergence is stated relating sequential epi- convergence in the weak topology to a certain graph convergence of the subdifferential operators. In the applications convergence of spectra of elliptic operators and convergence of semigroups are studied.

Altogether, the book is a valuable contribution to the limit analysis of variational problems and highly recommended to those interested in that area.

The book consists of three chapters. Chapter 1 introduces epi-convergence in a general topological setting and states its variational properties. Applications cover such difficult-to-handle limit processes as homogenization problems and singularly perturbed optimal control problems. Further, epi-convergence is geometrically interpreted as set- convergence of epigraphs and its relation to the concept of \(\Gamma\)- convergence, developed by De Giorgi’s Italian school, is pointed out.

In Chapter 2 topological properties of epi-convergence such as lower semicontinuity of epi-limits are studied, compactness results are given and epi-limits are compared for different topologies. In particular, it is shown that the epi-limit of a sequence of functions can be characterized as pointwise limit of their Moreau-Yosida approximates. Applications are taken from variational inequalities with varying obstacles, optimal control design and stochastic homogenization.

In Chapter 3 epi-convergence of convex functions on reflexive Banach spaces is studied with special emphasis on its relation to the Mosco- convergence and to the graph-convergence of subdifferential operators. It is shown that epi-convergence of a sequence of convex functions with respect to the weak topology is equivalent to the epi-convergence of their Yound-Fenchel transformations in the strong topology. From that result it is easily revealed that Mosco-convergence is equivalent to epi- convergence for both strong and weak topologies. The results are applied to a singularly perturbed optimal control problem and to the limit analysis for variational inequalities with obstacle constraints. Concerning subdifferential operators it is proved by means of Moreau- Yosida approximation that Mosco-convergence of a sequence of closed convex proper functions is equivalent to pointwise convergence of the resolvents of the corresponding subdifferential operators. A similar result in terms of epi-convergence is stated relating sequential epi- convergence in the weak topology to a certain graph convergence of the subdifferential operators. In the applications convergence of spectra of elliptic operators and convergence of semigroups are studied.

Altogether, the book is a valuable contribution to the limit analysis of variational problems and highly recommended to those interested in that area.

Reviewer: R.H.W.Hoppe

### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

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

35A15 | Variational methods applied to PDEs |

35B37 | PDE in connection with control problems (MSC2000) |

35B40 | Asymptotic behavior of solutions to PDEs |

49J27 | Existence theories for problems in abstract spaces |

49J40 | Variational inequalities |

49J50 | FrĂ©chet and Gateaux differentiability in optimization |

49N15 | Duality theory (optimization) |

49K40 | Sensitivity, stability, well-posedness |

90C30 | Nonlinear programming |