Most of the material of this book comes from graduate level courses on variational analysis, P.D.E., and optimization which have been given during the last decades by the authors: H. Attouch and G. Michaille at the University of Montpellier (France), G. Buttazzo at the University of Pisa (Italy). Our objective is twofold: The first objective is to provide to students the basic tools and methods of variational analysis and optimization in infinite dimensional spaces together with applications to classical P.D.E. problems. This corresponds to the first part of the book, from chapter 1 to 8, and which takes place in classical Sobolev spaces. We have made an effort to provide, as much as possible, a self-contained exposition, and try to introduce each new development from various perspectives (historical, numerical,…). The second objective, which is more oriented towards research, is to present new trends in variational analysis and some of the most recent developments and applications. This corresponds to the second part of the book, from chapter 9 to 15, where in particular are introduced the $BV(\Omega)$ spaces. This organization is intended to make the book accessible to a large audience, from students to researchers, with various backgrounds in mathematics, as well as physicists, engineers… As a guideline, we have been trying to portray direct methods in modern variational analysis. This is just like an anniversary, one century after D. Hilbert delineated them in his famous lecture at Coll\`ege de France, Paris, 1900. The extraordinary success of these methods is intimately linked with the development, all along the $20^{th}$ century, of new branches in mathematics: functional analysis, measure theory, numerical analysis, (nonlinear) P.D.E., optimization... We try to show in this book the interplay between all these theories, and also between theory and applications. Variational methods have proved to be very flexible. In recent years, they have been developped in order to study a number of advanced problems of modern technology like composite material, phase transitions, thin structures, large deformations, fissures, shape optimization... To grasp these often involved phenomena, the classical framework of variational analysis, which is presented in the first part, has to be enlarged. This is the motivation for the introduction in the second part of the book of some advances technics, like $BV$ and $SBV$ spaces, Young measures, $\Gamma$-convergence, recession analysis, relaxation methods… Finally, we wish to stress that variational analysis is a remarkable example of international collaboration. Quite all mathematical schools have contributed to its success and it is just like a modest symbol that this book has been written in collaboration between mathematicians of two of them, french and italian. This book owes much to the support of the Universities of Montpellier (France) and Pisa (Italy), of their Mathematical departments, and of the convention of cooperation which connects them.

### Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (Second Edition)

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*BUTTAZZO, GIUSEPPE;*

##### 2014

#### Abstract

Most of the material of this book comes from graduate level courses on variational analysis, P.D.E., and optimization which have been given during the last decades by the authors: H. Attouch and G. Michaille at the University of Montpellier (France), G. Buttazzo at the University of Pisa (Italy). Our objective is twofold: The first objective is to provide to students the basic tools and methods of variational analysis and optimization in infinite dimensional spaces together with applications to classical P.D.E. problems. This corresponds to the first part of the book, from chapter 1 to 8, and which takes place in classical Sobolev spaces. We have made an effort to provide, as much as possible, a self-contained exposition, and try to introduce each new development from various perspectives (historical, numerical,…). The second objective, which is more oriented towards research, is to present new trends in variational analysis and some of the most recent developments and applications. This corresponds to the second part of the book, from chapter 9 to 15, where in particular are introduced the $BV(\Omega)$ spaces. This organization is intended to make the book accessible to a large audience, from students to researchers, with various backgrounds in mathematics, as well as physicists, engineers… As a guideline, we have been trying to portray direct methods in modern variational analysis. This is just like an anniversary, one century after D. Hilbert delineated them in his famous lecture at Coll\`ege de France, Paris, 1900. The extraordinary success of these methods is intimately linked with the development, all along the $20^{th}$ century, of new branches in mathematics: functional analysis, measure theory, numerical analysis, (nonlinear) P.D.E., optimization... We try to show in this book the interplay between all these theories, and also between theory and applications. Variational methods have proved to be very flexible. In recent years, they have been developped in order to study a number of advanced problems of modern technology like composite material, phase transitions, thin structures, large deformations, fissures, shape optimization... To grasp these often involved phenomena, the classical framework of variational analysis, which is presented in the first part, has to be enlarged. This is the motivation for the introduction in the second part of the book of some advances technics, like $BV$ and $SBV$ spaces, Young measures, $\Gamma$-convergence, recession analysis, relaxation methods… Finally, we wish to stress that variational analysis is a remarkable example of international collaboration. Quite all mathematical schools have contributed to its success and it is just like a modest symbol that this book has been written in collaboration between mathematicians of two of them, french and italian. This book owes much to the support of the Universities of Montpellier (France) and Pisa (Italy), of their Mathematical departments, and of the convention of cooperation which connects them.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.