Chapter 3 Evans Pde Solutions.zip EXCLUSIVE

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Chapter 1: This chapter surveys the principal theoretical issues concerning the solving of partial differential equations. Evans introduces some common notation (such as partial differentiation) and discusses examples of commonly studied partial differential equations. He then discusses strategies of studying different PDEs and introduces functional analysis as a method of studying existence and uniqueness results for partial differential equations.

Both inhomogenuous and homogenuous solutions are discussed and Evans emphasizes the importance of finding explicit solutions through inequalities, integration by parts, Green's formula, convolution, energy methods, maximum principles, etc. (which are the backbone of finding explicit solutions). This chapter is very useful in that it is one of the rare occasions where one can find explicit formulas for PDEs.

Chapter 4: This chapter is a standalone chapter that contains a wide variety of techniques that are sometimes useful for finding explicit solutions to partial differential equations. Evans goes through detailed examples using separation of variables, similarity solutions, transformational methods, converting nonlinear PDE to linear PDE, and both asymptotics and power series. It is nice that Evans shows how to solve some nonlinear PDEs with techniques from linear PDEs. This chapter could be skipped on a first read-through.

Chapter 5: In my opinion, this is one of the most important chapters in the book. Evans dives into the more theoretical aspects of linear partial differential equations. He starts by introducing both Hölder and Sobolev spaces and then goes through local and global approximation theory through smooth functions (through the use of mollifiers). Next, he discussions the extension and trace theorems. In section $5.6$, Evans lays out crucial information about the embeddings of Sobolev spaces into others (what is known as Sobolev inequalities). He later discusses the Poincaré inequality. I would spend some time reading this chapter slowly (and do the exercises at the end of the chapter).

Chapter 6: This chapter is a generalization of Chapter $2$ (and the Laplace equation) with the new tools introduced in Chapter $5$. Evans discusses (and motivates) the existence weak solutions by the Lax-Milgram theorem, energy estimates, the Fredholm alternative, and applications of the Rellich-Kondrachov compactness theorem. In section $6.3$, Evans presents the regularity problem for weak solutions - Is the weak solution $u$ of the PDE

Chapter 7: This chapter is a natural extension of Chapter $6$ and covers linear partial differential equations which involve time. These are known as evolution equations where the solution evolves in time from a given initial configuration. Here, Evans gives a nice description of how to apply energy methods to general second-order parabolic and hyperbolic equations. The final subsection gives an outstanding introduction to the abstract theory of semigroups and describes contraction semigroups, the resolvent, and the Hille-Yosida theorem. There is also a nice application of semigroup theory to second-order parabolic PDEs.

Chapter 8: This chapter gives a wonderful introduction to the calculus of variations (you may also want to look at the book by Gelfand and Fomin). These are an important class of nonlinear problems which can be solved through using straightforward techniques from nonlinear functional analysis. Evans discusses the Euler-Lagrange equations, coercivity, lower semicontinuity, and the existence/uniqueness of minimizers. He then motivates the definition of the weak solution and shows how to find weak solutions to the Euler-Lagrange equations. In section $8.3$, the regularity theorems are discussed (which follow similarly from the analysis done in chapter $6$). The final subsections given an overview of various constraints, incomprehensibility, and critical points. The famous Moutain Pass theorem is discussed in section $8.5.1$ which is followed by Noether's theorem in section $8.6$.

Chapter 9: In the same spirit as chapter $4$, this is a collection of various techniques for proving the existence, nonexistence, uniqueness, and other properties of solutions to nonlinear elliptic and parabolic differential equations. The familiar fixed point theorems are discussed and Evans discusses direct applications of Banach's Fixed Point Theorem. Supersolutions, subsolutions, and the nonexistence of solutions are discussed in the context of nonlinear partial differential equations. The final subsection gives a nice overview of gradient flows as an extension of the abstract semigroup theory developed in section $7.4$. As with chapter $4$, this chapter could be skipped on a first reading.

Chapter 10: This chapter is devoted to the study of Hamilton-Jacobi equations (which are a modern application of PDE theory to the study of optimal control problems). As usual, Evans gives a good motivation for the definition of a viscosity solution and then discusses the consistency of solutions. Section $10.2$ shows uniqueness of viscosity solutions to Halmilton-Jacobi PDEs with initial values. The final section gives a good introduction to dynamic programming and control theory.

Chapter 11: This chapter is an abstract generalization of the later half of chapter $3$. Evans discusses systems of nonlinear, divergence structure first-order hyperbolic PDE, which arise as models of conservation laws. He goes through the familiar conservation and mass, momentum, and energy and then later demonstrates how to obtain weak solutions. Properties of traveling waves, rarefaction waves, and shock waves are discussed (with some good pictures modeling the physical parameters). The later sections include theory about how to attack systems of $2$ or more conservation laws and information about entropy conditions.

Chapter 12: This is a fun chapter about nonlinear wave equations. Evans discusses the semilinear wave equation, quasilinear wave equation, and how to think about the conservation of energy (in the spirit of the energy functional described in section 8.6.2). A bunch of generalizations of linear wave equations are discussed: existence of solutions, Sobolev inequalities, energy estimates, and nonexistence of solutions.

Chapters 1-4: These show how to find explicit solutions to linear PDEs. This material is covered in most standard PDE I/II courses. It is useful to understand how to apply Green's formula, integration by parts, inequalities, etc. as you will be doing a lot of this in the later chapters (and while reading PDE papers).

Chapters 5-8: These are the "bread and butter" of the book. Chapter $5$ demonstrates how use the Sobolev inequalities and chapter $8$ gives a bunch of different applications of the Euler-Lagrange equations. I would read these slowly and do the exercises at the end (I would suspect that these would be a "PDE 2" course in many universities).

Chapters 9-12: These are more advanced topics and Chapter $10$ gives a nice overview of some of optimal control theory. I like the presentation in chapter $12$ as it generalizes what is discussed in chapters $3$ and $6$.

As you are interested in the smoothness of solutions, I would review the regularity theorems presented in chapters $5$-$7$. You could then turn to other sources once you have mastered most of the exercises.

Back to the integral solution $u$ of the conservation law (the definition is somewhere earlier in that conservation law chapter of Evans):$\int^{\infty}_0 \int^{\infty}_{-\infty} \Big(u v_t + F(u) v_x\Big)\,dxdt + \int^{\infty}_{-\infty} gv\,dx \big|_{t=0}= 0,\tag{2}$for $v\in C^{\infty}_c(\mathbb{R}\times [0,\infty))$. Now $u$ only lies in $L^{\infty}\cap C_c$ ($u$ may not be differentiable anymore, thinking all those blows up in time, and shock waves in space!), the trick in (1) is not applicable anymore, here the way to prove this is to choose proper test function $v$.

In Evans' PDE book, chapter 3.2 (which you can consult here), it is presented the method of characteristic to solve the first order PDE\begin{cases}F(Du,u,x)=0 &\text{in }U \subset \mathbb R^n\\u=g &\text{on }\Gamma,\end{cases}(where $\Gamma \subseteq \partial U$ and $g:\Gamma \to \mathbb R$ are given) by converting the PDE into a system of ODE. After deriving the characteristic equations, it is introduced the flattening of the boundary so that the initial problem can be solved in a neighborhood of the initial datum $x^0 \in \Gamma$. Then the compatibility conditions are derived for a triple $(p^0,z^0,x^0)$ of initial conditions, and the compatibility conditions (equations (33) and (34) in the pdf linked above) are derived also for points near the said triple. Then we have the following lemma

Class meets MWF 12:00-12:50 in CB 347. The main goal of this course is to cover ordinary differential equations, linear algebra, and Strum-Liouville theory. The course continues with MA 507 in the spring of 2016. That course will cover complex variable theory and partial differential equations. NEW: You can get a pdf version of the 7th edition of Arfken on Science Direct. You simply new to log on using your link blue id. Midterm exam target date Wednesday, 21 October 2015, in CB 347.Final exam on Monday, 14 December 2015, 3:30--5:30 in CB 347.Notes on Exponentials, Logs, and Complex Numbers-everything you need to know!Notes by Professor Perry on linear algebra. We'll be discussing these during the last two weeks of class.Problem set 1 Due in class, Friday, 4 September 2015. Solutions to Problem set 1. Problem set 2 Due in class on Monday, 14 September. This is a short PS. Solutions to Problem set 2. Problem set 3 Due in class on Friday, 18 September 2015.Solutions to Problem set 3. Problem set 4 Due in class Monday, 28 September 2015. Solutions to Problem set 4. Problem set 5. Due in class Wednesday, 7 October 2015. You do not need to do 8.2.2 and 8.2.5 as we have not covered this material yet.Solutions to Problem set 5. Solutions to Test 1. Problem set 6. Due in class Friday, 16 October 2015.Solutions to Problem set 6. Problem set 7. Due in class, Monday, 2 November 2016.Solutions to Problem set 7. Problem set 8. Due in class, Monday, 9 November 2016.Solutions to Problem set 8. Problem set 9. Due in class Friday, 20 November 2016.Solutions to Problem set 9. Problem set 10. Due in class Monday, 7 December 2015.Solutions to Problem set 10. Solutions to the Final Exam. Have a good break! MA 113 Calculus I sections 025, 026, 027, 028, 029, 030This is the first semester of calculus. After a review of functions, we will study the derivative and integral. ALL SIX SECTIONS MEET FOR LECTURES: MWF 2:00-2:50 in CP 153. Final Exam Review 1 , Monday, 14 December 2015, 4:00--5:30 PM , KAS 213 FINAL EXAM, Tuesday, 15 December, 6--8 PM in BS 107 (same room as all the tests). Official policy of conflicts: ``Any student with more than two final examinations scheduled on any one date shall be entitled to have the examination for the class with the highest catalog number rescheduled.'' Email me by Monday, 7 December, if you have more than 2 exams on 15 Dec. or a conflict. This is my specific Course Syllabus for my sections of MA 113, sections 025, 026, 027, 028, 029, and 030, Fall 2015. It includes information on our TAs office hours and contact information.The general MA 113 course syllabus for fall 2015 is posted here. You will find the recitation work sheets, the course calendar, how to use Web Work, etc.Course Material for Fall 2014:MA 575 Introduction to AnalysisThis graduate class meets MWF 11:00-11:50 AM in CB 339. The goal of this course is to provide everyone with a firm foundation in the theory of functions of a single real variable. Although a lot of this is familiarfrom the calculus, we�ll carefully and rigorously study properties of functions,like continuity and differentiability, and the Riemann integral. We�ll also look atquestions of convergence of sequences of numbers and functions, including theimportant topic of uniform convergence of functions. These are fundamentalideas that all mathematicians should be familiar with. No class the week of November 3. This is the Course Syllabus for MA 575, Fall 2014.We will have a weekly problem session for discussing the problem sets on Friday afternoon at 3 PM. Let's meet in CB 339.NEW: make-up class on 14 and 21 November at this time in CB 339.Solutions to Test 1. Homework problem sets. Problem set 9 due Friday, 5 December.Exam topics: Series of functions, including power series; calculus of functions: continuity, uniform continuity, differentiation, Riemann integration; uniform continuity; basic point set topology. The final is on Wed. 17 Dec. 3:30-5:30 PM in CB 339. Good luck!Solutions to the final exam.MA 114 Honors Calculus II sections 009 and 010This is the honors Calculus II class for fall 2014 semester. Both sections 009 and 010 meet for lecturesMWF 1:00-1:50 in CB 110. Note change in room! This is my specific Course Syllabus for my honors sections of MA 114, sections 009 and 010, Fall 2014. Exam 3: Tuesday, 18 November, 5-7 PM, in BS 116 (Biological Sciences). General review: Monday, 17 November, CP 155 (Chemistry-Physics). The general MA 114 course syllabus for Fall 2014 is posted here. You will find the some of the recitation work sheets, old exams, the course calendar for the usual MA 114, how to use Web Work, etc. on these pages.Mr. George Lytle's web page and syllabus for the recitation sections. The worksheets for our recitations may be found here. Recitation for 009 mets in CB 341 and for 010 in CB 338. Note the change!CALENDAR: 29 Aug. We are finishing section 10.1 and will begin 10.2 on Wed. 03 Sept. Discussed limit laws and bounded monotone seq. Fri. 05 Sept. We did series and the principle of induction, will start 10.3 on Monday. Mon. 08 Sept. we did 10.3 on positive series, we'll begin 10.4 on Wed. Wed. 09 Sept. We did the alt series test. WW dates through WWH06 have been adjusted. Friday, 19 Sept. Finished power series, 10.6. First Exam on Tuesday, 23 Sept. 5-7 PM 116 BS.Wed. 24 Sept. Beginning 10.7 on Taylor's series. Fri. 26 Sept. Taylor polynomials and remainder, sec. 8.4. Mon. 29 Sept. Finished Taylor polynomials and began sec. 6.2 on computing volumes. Wed. 1 Oct. We did some volume calculations and the density. We'll begin 6.3 on Friday. We are doing volumes using the disk, washer and shell methods through Mon. Oct. 6. Work on WWh11-14. On Oct. 8 we did work, sec. 6.5. Week of 13 Oct.: We are now working in chapter 7 doing integration by parts, 7.1, and trig integrals, 7.2. Exam 2: Tuesday, 21 Oct. 5-7 PM in 116 BS. Review 20 Oct. 8 PM CP 155. Material from Taylor series through trig integrals (no trig substitutions). WWH-15 through 18 are now ready. Solutions to Test 2 are posted. Midterm grades based on tests 1 and 2 are posted. Wed. 29 Oct. We finished sec. 3.9 and 7.4 on hyperbolic trig functions. We began sec. 7.8 on numerical methods. 31 Oct. We finished our discussion of numerical approximation. WA2 is posted and due Friday, 14 Nov. We'll be doing section 8.1,8.3, and 9.1 next week. 12 Nov. We're finishing sections 9.1, 9.2, 9.3, and 9.4 on differential equations. Review for test 3 on Monday, 17 Nov. 8PM in CP 155. The Euler method and the logistic equation will NOT be on Test 3. 19 Nov. We did the logistic ODE, sec. 9.4. We'll finish 9.5 on Friday, and begin sections 11.1 and 11.2 on curves. Week of December 1: We finished parametrically defined curves, and are doing polar coordinates, sections 11.3 and 11.4. Final Review: Friday, 12 Dec. 7 PM in CB 106; Final Exam: Tuesday, 16 December, 6-8 PM in BS 116. The solutions to the final are posted below. Happy new year and Good luck during spring 2015 semester. Exam topics: trig integrals and trig substitution; integration by parts and partial fractions; hyperbolic trig functions (basic results); parameterized curves: slope function, arc length, area of a surface of revolution, speed; polar coordinates and polar curves (including arc length and area); differential equations: integrating factor, method of separation of variables, logistic equation and equilibrium solutions, slope fields; Taylor series and power series (radius of convergence and endpoint check). The won't be any proofs. The final is on Tues. 16 Dec. 6-8 PM in BS 116. Good luck!Solutions to Test 1.Solutions to Test 2.Solutions to Test 3.Solutions to the Final Exam.The first written assignment. you are to complete this on your own (you may discuss it with friends, me, and George). CORRECTION: Part b of problem 2: the error should be 10^{-6} mc^2, multiple by the rest energy. This way, the mc^2 factors will cancel out.. It is due on Friday, 17 October, at the beginning of class.Solutions to written assignment 1.The second written assignment. You are to complete this on your own (you may discuss it with friends, me, and George). . It is due on Friday, 14 November, at the beginning of class.Solutions to written assignment 2.The third written assignment. You are to complete this on your own (you may discuss it with friends, me, and George). . It is due on Friday, 5 December, at the beginning of class.Solutions to written assignment 3. There will not be a fourth written assignment.Course Material for Spring 2014:MA 641 Differential GeometryThis course will cover the basics of Riemannian geometry following the book Riemannian geometry by do Carmo (Birkhauser, 1988). We will study differentiable manifolds, Riemannian metrics, tangent and cotangent spaces, vector fields, geodesics, connections, and curvature. We will develop enough machinery to describe the spaces of constant curvature and complete manifolds. Riemannian geometry is built on the classical theory of curves and surfaces in space. It is recommended that the students look at a book such as Differential geometry of curves and surfaces by do Carmo to see the origins of the subject. NEW: Classroom change to CB 341!! 2b1af7f3a8