Andrea Braides
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.001.0001
- Subject:
- Mathematics, Applied Mathematics
This book introduces the main concepts of the theory of De Giorgi's Gamma-convergence and gives a description of its main applications to the study of asymptotic variational problems. The content is ...
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This book introduces the main concepts of the theory of De Giorgi's Gamma-convergence and gives a description of its main applications to the study of asymptotic variational problems. The content is based on results obtained during thirty years of research. The book is divided into sixteen short chapters, an Introduction, and an Appendix. After explaining how a notion of variational convergence arises naturally from the study of the asymptotic behaviour of variational problems, the Introduction presents a number of examples that show how diversified the applications of this notion may be. The first chapter covers the abstract theory of Gamma-convergence, including its links with lower semicontinuity and relaxation, and the fundamental results on the convergence of minimum problems. The following ten chapters are all set in a one-dimensional framework to illustrate the main issues of convergence without the burden of high-dimensional technicalities. These include variational problems in Sobolev spaces, in particular homogenization theory, limits of discrete systems, segmentation and phase-transition problems, free-discontinuity problems and their approximation, etc. Chapters 12-15 are devoted to problems in a higher-dimensional setting, showing how some one-dimensional reasoning may be extended, if properly formulated, to a more general setting, and how some concepts already introduced can be integrated with vectorial issues. The final chapter introduces the more general and abstract localization methods of Gamma-convergence. All chapters are complemented by a guide to the literature, and a short description of extensions and developments.Less
This book introduces the main concepts of the theory of De Giorgi's Gamma-convergence and gives a description of its main applications to the study of asymptotic variational problems. The content is based on results obtained during thirty years of research. The book is divided into sixteen short chapters, an Introduction, and an Appendix. After explaining how a notion of variational convergence arises naturally from the study of the asymptotic behaviour of variational problems, the Introduction presents a number of examples that show how diversified the applications of this notion may be. The first chapter covers the abstract theory of Gamma-convergence, including its links with lower semicontinuity and relaxation, and the fundamental results on the convergence of minimum problems. The following ten chapters are all set in a one-dimensional framework to illustrate the main issues of convergence without the burden of high-dimensional technicalities. These include variational problems in Sobolev spaces, in particular homogenization theory, limits of discrete systems, segmentation and phase-transition problems, free-discontinuity problems and their approximation, etc. Chapters 12-15 are devoted to problems in a higher-dimensional setting, showing how some one-dimensional reasoning may be extended, if properly formulated, to a more general setting, and how some concepts already introduced can be integrated with vectorial issues. The final chapter introduces the more general and abstract localization methods of Gamma-convergence. All chapters are complemented by a guide to the literature, and a short description of extensions and developments.
Andrea Braides
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0002
- Subject:
- Mathematics, Applied Mathematics
This chapter introduces all the abstract notions and results on Gamma-convergence. Starting from upper and lower-semicontinuous functions, Gamma-convergence is defined and its differences from other ...
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This chapter introduces all the abstract notions and results on Gamma-convergence. Starting from upper and lower-semicontinuous functions, Gamma-convergence is defined and its differences from other types of convergence are illustrated. A number of equivalent definitions are given, including the liminf inequality and the existence of recovery sequences; the equality of upper and lower bonds, through Moreau-Yosida transforms; and the convergence of minimum problems. Upper and lower Gamma-limits are introduced, and their properties in the framework of lower-semicontinuous functions are illustrated. The main properties of compactness and stability are proved. The notion of development by Gamma-convergence is introduced.Less
This chapter introduces all the abstract notions and results on Gamma-convergence. Starting from upper and lower-semicontinuous functions, Gamma-convergence is defined and its differences from other types of convergence are illustrated. A number of equivalent definitions are given, including the liminf inequality and the existence of recovery sequences; the equality of upper and lower bonds, through Moreau-Yosida transforms; and the convergence of minimum problems. Upper and lower Gamma-limits are introduced, and their properties in the framework of lower-semicontinuous functions are illustrated. The main properties of compactness and stability are proved. The notion of development by Gamma-convergence is introduced.
Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0007
- Subject:
- Mathematics, Analysis
This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where ...
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This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where certain functions that change during the process have values close to their infima. Like many other recursive constructions, the choice has a natural description using the language of infinite two-player games with perfect information. The chapter first considers the perturbation game used in Theorem 7.2.1 to formulate an abstract version of the variational principle before showing how to specialize it to more standard formulations. It then examines the bimetric variant of the smooth variational principle, along with the perturbation functions that are relatively simple. It concludes with an assessment of cases when completeness and lower semicontinuity hold only in a bimetric sense.Less
This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where certain functions that change during the process have values close to their infima. Like many other recursive constructions, the choice has a natural description using the language of infinite two-player games with perfect information. The chapter first considers the perturbation game used in Theorem 7.2.1 to formulate an abstract version of the variational principle before showing how to specialize it to more standard formulations. It then examines the bimetric variant of the smooth variational principle, along with the perturbation functions that are relatively simple. It concludes with an assessment of cases when completeness and lower semicontinuity hold only in a bimetric sense.
Peter Scholze and Jared Weinstein
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0022
- Subject:
- Mathematics, Geometry / Topology
This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important ...
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This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important foundational theorems about vector bundles on the Fargues-Fontaine curve. The first is the semicontinuity of the Newton polygon. The second theorem of Kedlaya-Liu concerns the open locus where the Newton polygon is constant 0. For the applications to the moduli spaces of shtukas, one needs to generalize the results to the case of G-torsors for a general reductive group G. The chapter then identifies the classification of G-torsors. It also looks at the semicontinuity of the Newton point.Less
This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important foundational theorems about vector bundles on the Fargues-Fontaine curve. The first is the semicontinuity of the Newton polygon. The second theorem of Kedlaya-Liu concerns the open locus where the Newton polygon is constant 0. For the applications to the moduli spaces of shtukas, one needs to generalize the results to the case of G-torsors for a general reductive group G. The chapter then identifies the classification of G-torsors. It also looks at the semicontinuity of the Newton point.