This is the third quarter of the year long sequence An introduction to PDEs--Math231 ABC. Even though having taken Math231 AB is NOT a prerequisite, the course does assume some background knowledge of some basics of the Sobolev space, elliptic, parabolic and hyperbolic PDEs. The Chapters 5, 6, 7 of Evans' book (Partial Differential Equations, AMS graduate studies in mathematics vol 19) are sufficient. The plan for this quarter is to cover Chapter 8 of the above mentioned Evans book, and give an introduction to the Monge-Ampère equation.
The first part of the course is on the topic of Calculus of Variations. A good introduction to the subject, particularly the calculus of variations of one variable, is given in the Dover book by Gelfand and Fomin (1963). This is highly recommended for the reading outside the classroom. Besides motivations and historic aspects of the subject it covers the basic problems in the case of one variable to a certain depth. Hence it is helpful to gain a more global understanding of the content of this part of course.
The importance of the subject is due to that the variational principles are essentially a manifestation of very general physical laws, which are valid in diverse branches of physics, ranging from classical mechanics to the theory of elementary particles. Our focus is on the analytic issue of the subject, particularly the related PDEs (so-called Euler-Lagrange equations) and the weak solutions which can be obtained by the considerations of the calculus of variations (via the minimizing sequences and the compactness). We shall also discuss the issue of regularity, the group invariance of the functional and related conservation laws. Besides Gelfand-Fomin's book, the book of Morrey (1966, Springer) and the book of Giaquinta (1983, Princeton Press) are also good sources to learn more about this topic. One can also find interesting history in the introduction of the book by Jost (Harmonic mappings between Riemannian manifolds).
The second part of the course is an introduction to the Monge-Ampère equation. The plan is to follow mainly the book by Gutièrrez (Chapters 1 and 4 of The Monge-Ampère equation, 2nd edition, 2016). The Monge-Ampère is an essential component of the Monge-Kantorovich transport problem; it is used in conformal, affine, Kähler geometries, in meteorology and in financial mathematics. The goal is to give a brief instruction to this fully nonlinear PDE. Notes shall be distributed if necessary. The books by Bakelman (Springer, 1994) and Pogorelov (Multidimensional Minkowski Problem, 1978) contains geometric aspects of the theory (mainly developed by Russian mathematicians). The paper by Caffarelli, Caffarelli-Nirenberg-Spruck, Calabi, Evans, Krylov-Safanov, Nirenberg, etc., and books of Gilbarg-Trudinger (Springer, 1983), Han-Lin (2011, 2nd edition, Courant Lecture Notes) are also wonderful sources.
Any mathematical proof in textbooks as put by A. Y. Khinchin `will undoubtedly seem very complicated to you. But it will take you only two to three week's work with pencil and paper to understand and digest it completely. It is by conquering difficulties of just this sort, that the mathematicians/or mathematical students grow and develop.' Even though none of the theorems involved in this course is a difficult one requiring the labor beyond several hours to digest, the same principle on the effort part applies. No good mathematics can be spoon-fed fast.
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Required Text:
Evans = Partial Differential Equations, by Evans, 2nd Ed., AMS, GSM, 2010
Figa= The Monge-Ampère equation and its applications, by A. Figalli., EMS, 2022
Gutièrrez = The Monge-Ampère equation, by Gutièrrez, 2nd Ed., Birkhäuser, 2016
Recommended other Texts:
Nonlinear Analysis on Manifolds-The Monge-Ampère equations, by T. Aubin, Springer, 1982
Elliptic Partial Differential Equations of Second Order, by Gilbarg and Trudinger, 2nd Ed., Springer, 1983
Elliptic Partial Differential Equations, by Han and Lin, 2nd Ed., Courant Lecture Notes in Mathematics, 2011
Partial Differential Equations, by J. Jost, 3rd Ed., Springer, 2013.
Monge-Ampère Equations of Elliptic Type (Noordhoof, 1964), Extrinsic Geometry of Convex Surfaces ( AMS, 1973), Multidimensional Minkowski Problem (Winston Sons, 1978), by A. V. Pogorelov
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