流形上的特征值问题英文版_毛井杜锋吴传喜.pdf

Eigenvalue Problems on Manifolds(流形上的特征值问题)Jing Mao Feng Du Chuan-Xi Wu(毛井杜锋吴传喜)山pSCIENCE PRESS三BeijingIn memory of Mr.Xu-Gui Mao,Prof.Jing Maosfather,who left us in October 2015 physically butwould stay with us mentally forever.PrefaceThe main content of this book is based on the Ph.D.thesis of Prof.Jing Mao(thefirst author of this book)submitted to University of Lisbon,Portugal and some ofhis published joint-works(on Spectral Geometry)with collaborators(for instance,the second and the third authors of this book,Prof.Isabel Salavssa,Prof.PedroFreitas,Prof.Q.-L.Wang,and so on).In the first four chapters,we mainly consider eigenvalue problems of some self-adjoint elliptic operators(e.g.,the Laplace operator,the drifting Laplacian,the p-Laplacian,the bi-harmonic operator,the bi-drifting Laplacian,the Paneitz operator,and so on),which appeared in the study of Physics,on bounded connected domains(with or without suitable boundary constraints).As we know,in the boundednesssetting,these operators have the discrete spectrum,and then one can try to estimateelements(i.e.,eigenvalues)in the discrete spectrum.Based on this fact,some newphysical isoperimetric inequalities,estimations for lower eigenvalues,and universalinequalities have been shown to readers.Meanwhile,some latest progresses andpossible questions desired to study have also been mentioned.Chapters 3 and 4were written by Prof.Feng Du.In December 2016,Prof.C.-X.Wu and Prof.JingMao have made careful check and necessary revisions to the first manuscript of thisbook.In the last chapter,based on the first authors research experience,we issueseveral open problems which might be interesting to readers from three aspects(i.e.,spectral problems concerning quantum strips and quantum layers,connectionbetween curvature flows and eigenvalue problem,and eigenvalue problem in Finslergeometry).Besides,some basic knowledge about warped products,which have beenused many times in this book,have been introduced in Appendix A.The authors would like to thank the financial support in part by NSF of China(Grant No.11401131),Key Laboratory of Applied Mathematics of Hubei Province,Faculty of Mathematics and Statistics of Hubei University.Jing MaoFaculty of Mathematics and Statistics of Hubei UniversityDecember,2016