### Partial Differential Equations

#### Spring Semester 2010/11

**Syllabus**

**1.** Nonlinear first-order PDE

* Complete integrals, envelopes
* Characteristics
* Introduction to Hamilton-Jacobi equations

**2.** The Cauchy-Kowalevski Theorem and Holmgren's Theorem

**3.** Laplace's equation

* Fundamental solution
* Mean-value formulas
* Properties of harmonic functions
* Green's function
* Energy methods

**4.** Heat Equation

* Fundamental solution
* Mean-value formula
* Properties of solutions
* Energy methods

**5.** Wave equation

* The Cauchy Problem
* The inhomogeneous equation
* Fourier analysis of the wave operator

**6.** Sobolev spaces

**7.** Second-order elliptic equations

* Lax-Milgram Theorem
* Fredholm alternative
* Application

**Main textbook**

**[E]** Evans, Lawrence C., Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

**Additional bibliography**

**[DB]** DiBenedetto, Emmanuele, Partial differential equations. Second edition. Cornerstones. Birkhäuser Boston, Inc., Boston, MA, 2010.

**[F]** Folland, Gerald B., Introduction to partial differential equations. Second edition. Princeton University Press, Princeton, NJ, 1995.

**[G]** Garabedian, Paul R., Partial differential equations. Second edition. Chelsea Publishing Co., New York, 1986.

**[J]** John, Fritz, Partial differential equations. Fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1982.

**Evaluation**

- Problems are due 2 weeks after they are assigned. A list of assigned problems will be kept on this webpage.

- The final grade will be determined as follows: 40%
of the grade obtained in the problems assigned during the semester and
60%
of the grade obtained in the final exam.

- There will be 2 dates for the final exam. If a student attends both of them, then the lower of the two grades will not be considered.

- For a student to pass the course he or she will have to obtain at least 8 out of 20 in one of the exams.

- A student with final grade greater than 17 will be asked to attend an oral exam. He or she may decline to do so, but in that case the final grade will be 17.

Summaries

**Problems assigned in class**

- Due March 9, 2011:
[J] p. 18 exercise 1(b) and 1(c), p. 19 exercise 6.

- Due March 30, 2011:
[J] p. 31 exercise 1(c), p. 31 exercise 2.

- Due April 20, 2011:
Solve u_t+u_x^2/2 = 0 with u(x,0) = - |x| using the Hopf-Lax formula. [E] p. 235 exercise 9.

- Due May 4, 2011:
[J] p. 102 exercises 3 and 4.

- Due May 17, 2011:
[E] p. 86 exercises 6, 9, 12 and 13.

- Due May 25, 2011:
[J] p. 132 exercises 2(b) and 3(a-b)

First Exam and
solutions

Second Exam and
solutions

**Final grades**

Época Especial: July 21, 11:30 am, room P2

#### Spring Semester 2009/10

Problem set grades

Summaries

First Exam and
solutions

Second Exam and
solutions

Exam grades