Localization Approaches in Strongly Indefinite Problems

Yanheng Ding, Tian Xu (Autoren)

Buch | Hardcover

281 Seiten

2025 | 2025 ed.
Springer Verlag, Singapore
978-981-97-9538-3 (ISBN)

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Several important problems arising in Physics, Differential Geometry and other topics lead to consider semilinear variational equations of strongly indefinite type and a great deal of work has been devoted to their study. From the mathematical point of view, the main interest relies on the fact that the tools of Nonlinear Functional Analysis, based on compactness arguments and non-degenerate structure, in general cannot be used, at least in a straightforward way, and some new techniques have to be developed.

This book discusses some new abstract methods together with their applications to several localization problems, whose common feature is to involve semilinear partial differential equations with a strongly indefinite structure. This book deals with a variety of partial differential equations, including nonlinear Dirac equation from quantum physics (which is of first order), coupled system of multi-component incongruent diffusion and spinorial Yamabe type equations on spin manifolds. The unified framework in this book covers not only the existence of solutions to these PDEs problems, but also asymptotic behaviors of these solutions. In particular, the results for the nonlinear Dirac equations show several concentration behaviors of semiclassical standing waves under the effect of external potentials and the results for the spinorial Yamabe type equations show the existence of conformal embeddings of the 2-sphere into Euclidean 3-space with prescribed mean curvature.

This book will be appealing to a variety of audiences including researchers, postdocs, and advanced graduate students who are interested in strongly indefinite problems.

Yanheng Ding, Professor of Jilin University and Academy of Mathematics and Systems Science, Chinese Academy of Science. His research interests include nonlinear functional analysis, critical point theory and its applications in PDEs. He received the Alexander von Humboldt research fellowship in 1996. Tian Xu, Professor of Department of Mathematics, Zhejiang Normal University. His research interests include variational methods for strongly indefinite problems, analytical problems in conformal geometry and its applications. He received the Alexander von Humboldt research fellowship in 2017.

Chapter 1. Variational Problems —– A Brief Retrospective.- Chapter 2. Strongly Indefinite Problems —– Examples and Motivations.- Chapter 3. Localized Energy Estimates for Strongly Indefinite Functionals.- Chapter 4. Semiclassical Standing Waves of Nonlinear Dirac Equations.- Chapter 5. Effect of External Potentials in a Coupled System Reaction-Diffusion.- Chapter 6. The Spinorial Brezis-Nirenberg Problem.- Chapter 7. Isometrically Embedded Sphere with Prescribed Mean Curvature.- Chapter 8. Further Problems with Strongly Indefinite Structures.

Erscheinungsdatum	16.05.2025
Reihe/Serie	Series in Contemporary Mathematics
Zusatzinfo	5 Illustrations, color; 5 Illustrations, black and white
Verlagsort	Singapore
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	critical point theory • localized analysis • minimax theorems • Nonlinear Dirac Equation • spinorial Yamabe type equations • strongly indefinite problems • variational methods
ISBN-10	981-97-9538-9 / 9819795389
ISBN-13	978-981-97-9538-3 / 9789819795383
Zustand	Neuware