# Math 110A: Introduction to PDE (Fall 2017)

### Discussions: Th 4:00-4:50pm (Sec A01, YL) or 5:00-5:50pm (Sec A02, YL) or 6:00-6:50pm (Sec A03, BC) in WLH 2206

 Instructor: Andrej Zlatos Office: APM 5121 Office Hours: Th 9:15-10:45am (in APM 5121) plus 10 min after each lecture (in HSS 1330) E-mail: xzlatosx@ucsd.edu (remove the two x) TAs: Yuwen Li (YL) and Benjamin Ciotti (BC) Office: AP&M 6414 (YL) and AP&M 5712 (BC) Office Hours: M 4-6pm + F 4-6pm (YL) and M 5-6:15pm + Th 5-5:45pm (BC) E-mail: xyul739x@ucsd.edu (YL) and xbciottix@ucsd.edu (BC) (remove the two x)

Textbook: Partial Differential Equations: An Introduction (Second Edition) by Walter Strauss (a copy is on reserve in the library). We will cover most of chapters 1-6 and a part of chapter 9, with some adjustments. Lecture notes containing everything as presented in class will be posted here.

Quiz: Tuesday 10/24 (in class)

Midterm exam: Tuesday 11/14 (in class)

Final exam: Wednesday 12/13 at 11:30am (in HSS 1330)

Grading: Homework 25%, Quiz 10%, Midterm exam 25%, Final exam 40%. At least 50% of homework points as well as a passing grade on the final exam must be earned in order to pass the course.

Announcements:

• Review the material from Appendices A.1-A.4 in the textbook.
• I will generally follow the textbook but there will be some differences in the selection of topics and how they are covered. I will post lecture notes on TritonEd that will contain everything from the class.
• Midterm review will be on Thursday 11/9 at 7-9pm in PETER 102.
• Midterm topics are everything covered up to and including 11/7 (i.e., up to section 4.1), with the exception of: classification of second order PDE (elliptic/parabolic/hyperbolic), formulas for solutions of the wave equation in 2 and 3 dimensions, divergence theorem and energy-based arguments in several dimensions, and the reflection method on bounded/finite intervals. You also do not need to remember the source parts of the formulas in sections 3.3 and 3.4 - if needed, these will be provided (you *do* need to remember the homogeneous parts of those formulas, which are from chapter 2).
• Final exam topics are everything covered in class with the exception of: divergence theorem and energy-based arguments in 2 or more space dimensions, Section 4.3, and the reflection method on bounded/finite intervals. Final exam review will be done in class.

Homework assignments (due dates in parentheses): While some of the problems below are technically proofs, they are mainly short arguments illustrating the use of some of the concepts taught in class.

 HW 1 Section 1.1: Exercises 3bcdh, 10, 11 (10/10) Section 1.3: Exercises 6, 7, 10 Section 1.4: Exercises 1, 4 Section 1.5: Exercises 1, 4 HW 2 Section 1.2: Exercises 1, 3, 6, 7, 10 (10/17) Section 2.1: Exercises 1, 5, 9 Section 2.2: Exercises 3, 5 HW 3 Section 2.3: Exercises 3, 4, 7, 8 (in 3a, 4a use the strong version of maximum principle mentioned on p. 42; in 7b assume instead $v(x, 0) \ge 0$) (10/24) Section 9.1: Exercises 2, 6a, 7 Section 9.2: Exercises 2, 3, 5 Extra problems for interested Math majors HW 4 Section 1.6: Exercises 1, 2, 6 (10/31) Section 2.4: Exercises 6, 9, 11, 16, 18 Section 2.5: Exercise 2 Problem 10. Assume that $\varphi\in C(\mathbb R^N)$ and $I=\int_{\mathbb R^N} |\varphi(x)|\,dx$ is finite (i.e., $\varphi$ is integrable), and let $u(x,t)= (4\pi \kappa t)^{-N/2} \int_{\mathbb R^N} e^{-|x-y|^2/4\kappa t}\varphi(y)\,dy$. Show that for each $\epsilon>0$ there is $T>0$ such that $|u(x,t)|<\epsilon$ for all $(x,t)\in\mathbb R^N\times[T,\infty)$. (That is, any solution of the heat equation on $\mathbb R^N$ with an integrable initial condition converges uniformly to 0 as $t\to\infty$. Think about why this should be true from the physical perspective.) HW 5 Section 3.1: Exercises 1, 2 (11/7) Section 3.2: Exercises 3, 10 Section 3.3: Exercises 1, 3 Section 3.4: Exercises 3, 6, 13 Section 9.4: Exercise 3 (with initial condition $u(x,y,z,0)=\varphi(x,y,z)\in C_b(\mathbb R^2\times(0,\infty))$ on the half-space $\{z>0\}$) HW 6 Section 4.1: Exercises 2, 3, 4, 6 (11/21) Section 4.2: Exercises 2, 4 Section 4.3: Exercise 2 Problem 8. Let $D\subseteq\mathbb R^N$ be an open bounded domain with a $C^1$ boundary $\partial D$ and let $\Omega=D\times\mathbb R^+$. Consider the Schrodinger equation $u_t=i\Delta u$ on $\Omega$ for complex-valued functions $u$, with homogeneous Dirichlet or Neumann boundary condition on $\partial D\times\mathbb R^+$. Define energy $E(t)=\int_D |u(x,t)|^2 dx$ and show that for any solution $u\in C^{2,1}(\bar\Omega)$ it is conserved. Recall that since $u$ is complex valued, $|u|^2=u\bar u$. (Just as in Lecture 11, this can be used to prove uniqueness of $C^{2,1}(\bar\Omega)$ solutions to the IBVP with any initial condition $u(x,0)=\varphi(x)\in C^2(\bar D)$ because the PDE is linear.) Problem 9. Consider the setting of problem 8 but with equation $u_{tt}=\Delta u - u$ and $C^{2}(\bar\Omega)$ real-valued solutions. Find an "energy" for this equation that is conserved in time. (Hint: Find the time-derivative (in this problem) of the original energy for the wave equation from the proof of uniqueness. Then change this energy function in such a way that the time derivative of the new energy function becomes zero.) Problem 10. Find all $C^2([-1,0])$ solutions to BVP \begin{align*} u_{xx}=0 & \qquad \text{on $(-1,0)$} \\ u(-1)+u'(-1)=u(0)=0 & \end{align*} Why does the answer not contradict our uniqueness theorem for Laplace's equation? HW 7 Section 5.1: Exercises 2, 4, 5, 8, 9 (12/1) Section 5.2: Exercises 11, 14 Section 5.6: Exercise 4 with the second boundary condition being instead $u(l,t)=1$ Section 6.1: Exercises 2, 4 HW 8 Section 6.1: Exercises 5, 7, 11 (12/8) Section 6.2: Exercises 2, 4, 6, 7 (for 6, look at Example 2 in Section 6.2) Section 6.3: Exercises 1, 3 Problem 10. With $D$ the disc in $\mathbb R^2$ with radius 3 and centered at the origin, find all solutions to BVP \begin{align*} \Delta u = 0 & \qquad \text{on $D$} \\ \frac {\partial u}{\partial n}(3\cos\phi,3\sin\phi)= h(\phi) & \qquad \text{on $[-\pi,\pi]$} \end{align*} Write your answer in terms of a series involving the Fourier coefficients of $h$.

Exams and Quiz: You must bring a photo ID with you to the exams and to the quiz. All three will be taken with a pen and paper only; the use of books, notes, calculators, cell phones, etc. will not be permitted.

You must show your work and understanding of the material. A correct answer alone is sufficient only if the statement of the problem indicates so.

No make-up exams and quiz will be given. If a student misses a quiz or the midterm exam due to a sufficiently serious medical condition/emergency, (s)he needs to email me about this as soon as reasonably possible. Upon receiving satisfactory supporting documentation I will approve an alternate grading scheme for that student, with the final exam being worth an extra 10% or 25% (depending on which assignment was missed).

Homeworks: Homework will mostly be due on Tuesdays (the last two on Fridays) by 5:00pm in a drop-box in the basement of APM. No late homework will be accepted, except in cases of sufficiently serious medical conditions/emergencies.

Discussing any homework problem with other students taking the course is fine but only after you have thought about it and tried to solve it yourself for at least 10 minutes. Moreover, solutions must be written up individually, without help from others and without copying/paraphrasing their work or any other source. This approach will help you on the quiz and exams, which account for most of your course grade.

Use complete English+Math sentences, e.g., "If $a < b$, then $a+\sin a < b+\sin b$." Write clearly and concisely, do not hand in a rough draft or first attempt. Write each problem on a separate page and label it clearly. (You can use multiple pages for a problem if needed but do not include two problems on the same page; it is OK to use 2 sides of a single sheet for 2 distinct problems.) Write legibly and staple all sheets together in the correct order (folding corners of the sheets and/or taping them together is not the same as stapling).

Regrading policy: Homeworks, the quiz, and the midterm exam will be returned during the discussions. If you do not retrieve your assignment during the discussion, arrange to pick it up from your TA as soon as possible because any regrade requests will only be considered within one week of the date when the assignment was first made available for pickup (except in cases when this is not possible due to medical or other relevant reasons). Moreover, quiz and midterm regrade requests must be made when you pick up the assignment from the TA, before you leave the room. If upon receiving a regraded assignment you still think it is graded incorrectly, discuss this with your TA. If a disagreement persists, ask the TA to give me the assignment for review.

Recording of grades: It is your responsibility to check that your homework and exam scores are recorded correctly on TritonEd. If you find a discrepancy, you must notify your TA before the final exam and also present the original assignment. You should therefore keep all your graded assignments for future reference.

Communication with me and your TA: Office hours are the preferred venue for help/advice. If you cannot attend these, or in case of an "emergency" when there is no time to wait for office hours, e-mail us either about setting up an appointment or with the specific question. We will be happy to respond by email to questions which have short and straightforward answers. If, however, your question is long or requires more than a couple of lines to answer, you should see us in person as e-mail is frequently not a very efficient means to communicate about math. If you have a question about class policies, please read this page in full first.

Occasionally I will send course-related emails through the classlist server, or post announcements on this page. You are responsible for regularly checking your email account which receives such emails as well as this page.

Academic dishonesty (cheating): Academic dishonesty (including violations of most of the above rules) is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university (see here for more information). Any student who witnesses an instance of academic dishonesty should report this to the instructor or the TA.

I reserve the right to change any of the above if I find it necessary, in which case I will inform everyone about it well in advance.