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(Russian) Second edition, revised. Izdat. “Nauka”, Moscow, 1973. 576 pp. 8. Ireneo Peral, Multiplicity of solutions for the p-Laplacian. Int. center for theoretical physics, Trieste, 1997 9. The p-Harmonic Equation and Recent Advances in Analysis - Contemporary Mathematics 370, Ed. Pietro Poggi-Corradini, Kansas State University, Editor - AMS, 2005 10. M. Struwe, Variational methods with applications, Sec. Ed. Springer-Verlag, New York, 1993 Papers From 2000 on 1. W. Allegretto, Sturm theorems for degenerate elliptic equations. Proc. Amer. Math. Soc. 129:10 (2001),3031–3035. 2. A. Anane, N Tsouli, On a resonance condition between the first and the second eigenvalues for the p-Laplacian, Int. J. Math. Sci., 26:10 (2001) 625 - 634. 3. Aronsson, Gunnar; Crandall, Michael G.; Juutinen, Petri A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 439–505. 4. C. Azizieh, P. Clement, E. 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Juutinen, Petri; Lindqvist, Peter; Manfredi, Juan J., The infinity Laplacian: examples and observations. Papers on analysis, 207–217, Rep. Univ. Jyv¨askyl¨a Dep. Math. Stat., 83, Univ. Jyv¨askyl¨a, Jyv¨askyl¨a, 2001. 25. Lindqvist, Peter; Manfredi, Juan; Saksman, Eero, Superharmonicity of nonlinear ground states. Rev. Mat. Iberoamericana 16 (2000), no. 1, 17–28. 26. R. Manasevich, J. Mawhin, The spectrum of p-Laplacian systems under Dirichlet, Neumann and periodic boundary conditions. Morse theory, minimax theory and their applications to nonlinear differential equations, 201–216, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, 2003. 27. R. Manasevich, G. Sweers, A comparison result for perturbed radial p-Laplacians. J. Math. Anal. Appl. 291:1 (2004) 1–19. 28. P. Pucci, J. Serrin, The strong maximum principle revisited. J. Diff. Equa. 196:1 (2004), 1–66. 29. J. Serrin, H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189:1 (2002) 79–142. 30. P. Takaˇc, On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51:1 (2002) 187–237. 1990–1999 1. Acerbi, E.; Fusco, N. Local regularity for minimizers of nonconvex integrals., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), no. 4, 603–636 (1990) 2. W. Allegretto, Y. X. Huang, Principal eigenvalues and Sturm comparison via Picone’s identity. J. Diff. Equa. 156:2 (1999) 427–438. 3. Alvarez, O. Lasry, J.-M.; Lions, P.-L., Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76 (1997), no. 3, 265–288. 4. A. Ambrosetti, J. Garcia Azorero, I. Pearal Alonso, Multiplicity results for some nonlinear elliptic equations, J. Func. Anal. 137 ( 1996) 219-242. 5. A. Anane, N Tsouli, On the second eigenvalue of the p-Laplacian, Pitman Research Notes in Math. 343 (1996) 1-9. 6. G. Aronsson, On p−harmonic functions, convex duality and an asymptotic formula for injection mould filling. Euro. 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仅限学术研究使用，严禁商业用途，作者和出版社如有异议，我立即删除附件。如果觉得本书比较好，请购买正版图书。也欢迎各位博友讨论本书内容。欢迎学术交流。 Functional Analysis,Sobolev Spaces and Partial Differential Equations (Haim Brezis)(2011) Preface This book has its roots in a course I taught for many years at the University of Paris. It is intended for students who have a good background in real analysis (as expounded, for instance, in the textbooks of G. B. Folland , A. W. Knapp , and H. L. Royden ). I conceived a program mixing elements from two distinct worlds: functional analysis (FA) and partial differential equations (PDEs). The first part deals with abstract results in FA and operator theory. The second part concerns the study of spaces of functions (of one or more real variables) having specific differentiability properties: the celebrated Sobolev spaces, which lie at the heart of the modern theory of PDEs. I show how the abstract results from FA can be applied to solve PDEs. The Sobolev spaces occur in a wide range of questions, in both pure and applied mathematics. They appear in linear and nonlinear PDEs that arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, and physics. They belong to the toolbox of any graduate student in analysis. Unfortunately, FA and PDEs are often taught in separate courses, even though they are intimately connected. Many questions tackled in FA originated in PDEs (for a historical perspective, see, e.g., J. Dieudonn and H. BrezisF. Browder ). There is an abundance of books (even voluminous treatises) devoted to FA. There are also numerous textbooks dealing with PDEs. However, a synthetic presentation intended for graduate students is rare. and I have tried to fill this gap. Students who are often fascinated by the most abstract constructions in mathematics are usually attracted by the elegance of FA. On the other hand, they are repelled by the neverending PDE formulas with their countless subscripts. I have attempted to present a smooth transition from FA to PDEs by analyzing first the simple case of onedimensional PDEs (i.e., ODEsordinary differential equations), which looks much more manageable to the beginner. In this approach, I expound techniques that are possibly too sophisticated for ODEs, but which later become the cornerstones of the PDE theory. This layout makes it much easier for students to tackle elaborate higher-dimensional PDEs afterward. Functional Analysis,Sobolev Spaces and Partial Differential Equations (Haim Brezis)
明年4月4-8日，英国剑桥的 Isaac Newton 数学科学研究所和威尔士的数学与计算科学研究所(WIMCS)将联合在Swansea大学举办一个 PDE计算挑战的学术会议。具体信息如下： ---------------------------------------------------------------------------------------------------------------------- INI/WIMCS MEETING 2011 In April 2011, the Isaac Newton Institute for Mathematical Sciences (INI), Cambridge,and the Wales Instituteof Mathematical and Computational Sciences (WIMCS) will jointly organise a meeting on Computational Challenges in Partial Differential Equations at Swansea University. The meeting will start on Monday, April 4th and end on Friday, April 8th. This will be a follow-up meeting which, hopefully, will build upon the success of the six-month research programme, of the same name , that was held at the INI in 2003. Financial support for the meeting has been provided by the INI , WIMCS and the Centre for Numerical Analysis and Intelligent Software (NAIS). ORGANISERS The scientific organisers of the meeting are: Mark Ainsworth, University of Strathclyde Charles M. Elliott, University of Warwick Kenneth Morgan, Swansea University Endre Sli, University of Oxford MEETING FORMAT The meeting will consist of eight half-day sessions, each concentrating on a particular research area that is currently attracting significant interest within the community. The sessions will address the following themes: Multiscale modelling Interface modelling PDEs on surfaces and geometric evolution problems Biomedical applications, including new modelling techniques and patient-specific applications Computational rheology Atomistic-to-continuum passage, density functional theory and quasi-continuum methods Low order modelling: widening the range of high-fidelity time-dependent simulations Uncertainty modelling SPEAKERS Each half-day session will consist of invited presentations by four leading scientists. The following have agreed to participate and make presentations: A. Abdulle, EPFL, Switzerland S. Adhikari, Swansea University, UK J. W. Barrett, Imperial College London, UK S. Bartels, Universitt Bonn, Germany E. Cancs, CERMICS-ENPC, France S. J. Cox, Aberystwyth University, UK K. Deckelnick, Otto-von-Guericke-Universitt, Magdeburg, Germany Q. Du, Pennsylvania State University, USA G. Dziuk, Universitt Freiburg, Germany R. S. Elliott, University of Minnesota, USA C. Farhat, Stanford University, USA L. Formaggia, Politecnico di Milano, Italy T. Y. Hou, Caltech, USA G. E. Karniadakis, Brown University, USA C. Le Bris, CERMICS-ENPC, France T. Lelivre, CERMICS-ENPC, France P. Lin, University of Dundee, UK J. S. Lowengrub, UC Irvine, USA Y. Maday, Universit Pierre et Marie Curie, France K. Miller, The University of Western Australia, Australia G. S. Mishuris, Aberystwyth University, UK P. Nithiarasu, Swansea University, UK C. Ortner, University of Oxford, UK A. T. Patera, MIT, USA J. Peraire, MIT, USA D. Peric, Swansea University, UK T. N. Phillips, Cardiff University, UK W. Ren, Courant Institute, USA G. Rozza, EPFL, Switzerland S. Ruuth, Simon Fraser University, Canada S. Sherwin, Imperial College London, UK B. Stinner, University of Warwick, UK PROGRAMME An outline version of the programme has been prepared. This will be updated as more information becomes available. LOCATION The meeting will be held on the campus of Swansea University, in theFaraday Building. The location of this buildingis clearly marked on the campus map . Swanseacan be reached by road or by direct train service from London. The nearestmajor airport is Cardiff , with international connections via Amsterdam and Paris. REGISTRATION FEES The registration fee is 155. This covers attendance at the sessions, the welcome drinks reception on April 4th and the special dinner on April 7th. The costs of morning coffee, afternoon tea and lunches, for each day, are also included. To register, please complete and return the registration form . ACCOMMODATION A limited number of en-suite rooms, in student campus accommodation, are available. To reserve accommodation, please complete and return the accommodation form . CONTACT DETAILS Please address queries forfurther informationto k.morgan@swansea.ac.uk
Abstract. These are notes from a two-quarter class on PDEs that are heavily based on the book Partial Differential Equations by L. C. Evans, together with other sources that are mostly listed in the Bibliography. The notes cover roughly Chapter 2 and Chapters 5--7 in Evans. There is no claim to any originality in the notes, but I hope for some readers at least they will provide a useful supplement. Notes on Partial Differential Equations ( John K.Hunter ).pdf Notes on Partial Differential Equations ( John K.Hunter ).pdf Measure Theory ( John K. Hunter ).pdf
1. Variational Problems in SBV . 2. Critical Point Theorems and Applications to Nonlinear Differential Equations 3. Critical Exponents and Dimensions for Elliptic Equations 4. Variational Problems in the Space of Functions of Bounded Variation 5. Variational Techniques for Sturm-Liouville Eigenvalue problem 6. Variational Inequalities of Elliptic and Parabolic Type 7. Variational Methods for k-Hessian Equations 8. A local minimax-newton's method for finding critical points with symmetries 9. Lusternik-Schnirelman theory in partially ordered ordered Hilbert spaces
1. Notes on the p-Laplace equation . 2. Topics in Harmonic Analysis . 3. Lecures on Lipschitz analysis . 4. Nonlinear potential theory on metric spaces . 5. The Obstacle Problem . 6. Lectures on Regularity of free boundaries in Obstacle-type 7. Heat Method in Nonlinear Elliptic Equations
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