Localization Approaches in Strongly Indefinite Problems
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Beschreibung
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.
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.
Presents the latest developments of the critical point theory and its applications in nonlinear PDEs Unifies into a broad framework a wide variety of PDEs problems Explores state-of-the-art research at the interface among PDEs, differential geometry and calculus of variations
Autor*in
Yanheng Ding
Themen in »Localization Approaches in Strongly Indefinite Problems«
variational methods critical point theory strongly indefinite problems nonlinear Dirac equation spinorial Yamabe type equations minimax theorems localized analysis
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"This monograph makes a significant contribution to the intersection of nonlinear functional analysis, variational methods, and partial differential equations (PDEs). ... This book is an excellent supplement to the existing literature on strong indefinite variational problems. Both new and experienced researchers can use it to access the most up-to-date information on advanced techniques in the field." (Chun-Lei Tang, Mathematical Reviews, April, 2026)
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