Math 2900: Partial Differential Equations 1
Mon, Wed, Fri 12:00 - 12:50pm -- Thackeray 524
Instructor.
Dr. Marta Lewicka (office hours in Thackeray 408, Wednesday 5:45pm - 6:30pm)
Textbook:
"Partial Differential Equations" by Lawrence C. Evans. I will use the 1998 edition by the American Mathematical Society, Reprinted with corrections 1999.
Grades.
Your final grade will be based on homeworks. Homework problems should be solved independently and the usual code of conduct applies.
Core topics.
This course is a rigorous graduate level introduction to partial differential equations.
1. The first part of this course will study the three classical linear PDEs: Laplace's equation, the heat equation, and the wave equation.
2. Then we will move on to nonlinear PDE, and we will cover the method of characteristics, Hamilton-Jacobi equations, and conservation laws.
3. In the final part of the course we will study Sobolev spaces and general techniques for second-order elliptic equations.
4. Other topics we may touch on include Fourier and Laplace transform techniques, and separation of variables.
Calendar.
11 Jan (Mon): First Class
7, 9, 11 March (Mon, Wed, Fri): Spring Break - no classes
29 Apr (Fri): Last Class
Homework
Homework 1 (due January 22)
Section 1.5: problems 1, 2, 3. Section 2.5: problems 1, 2.
Homework 2 (due January 29)
1. Let $\phi\in L^1(U,\mathbb{R})$ be a Lebesgue-integrable function on some measurable set $U\subset\mathbb{R}^n$. Using only the definition of Lebesgue integral, prove the following: for every $\epsilon>0$ there exists $\delta>0$ such that every measurable set $A\subset U$ satisfying $|A|\leq \delta$ has the property: $\int_A |f| \leq \epsilon$.
2. Section 2.5: problems 3, 4, 5, 6.
Homework 3 (due February 17)
1. Let $g:\mathbb{R}^{n-1}\to\mathbb{R}$ be a continuous and bounded function. Show that the function $u(x) = \int_{\mathbb{R}^{n-1}} \frac{g(y)}{|y-x|^n}~\mbox{d}y$ is smooth in $\mathbb{R}^n_+$.
2. Let $g:\partial B(0,r)\to\mathbb{R}$ be a continuous function. For $x\in B(0,r)$ define: $u(x) = \int_{\partial B(0,r)} g(y) \frac{r^2 - |x|^2}{n |\omega_n| r |x-y|^n}~\mbox{d}S(y)$. Prove that $u$ is harmonic in $B(0,r)$ and that $\lim_{x\to x_0} u(x) = g(x_0)$ for every $x_0\in\partial B(0,r)$.
3. Let $p\geq 2$. Show that the function $u\in\mathcal{C}^2(\bar U)$ solves the problem: $$\mbox{div}\big(|\nabla u|^{p-2}\nabla u\big) = 0 \quad \mbox{ in } U, \qquad u=g \quad \mbox{ on } \partial U$$ if and only if $u$ minimizes the following energy: $I_p(v)=\int_U|\nabla v|^p$ among $\{v\in \mathcal{C}^2(\bar U); ~ v=g \mbox{ on } \partial U\}$.
4. Section 2.5: problems 9, 10.
Homework 4 (due February 29)
1. Prove that, given $\epsilon>0$, the fundamental solution to the heat equation is uniformly bounded with all its derivatives on $\mathbb{R}^n\times [\epsilon, \infty)$. Further, show that the function $u(x,t) = \frac{1}{t^{n/2}} \int_{\mathbb{R}^n}g(y) e^{-\frac{|x-y|^2}{4t}}~\mbox{d}y$ is $\mathcal{C}^\infty(\mathbb{R}^n\times [\epsilon, \infty))$ when $g$ is continuous and bounded on $\mathbb{R}^n$.
2. Let $\Phi(x,t)$ be the fundamental solution to the heat equation, defined on $\mathbb{R}^n\times [0,\infty)$. Consider the unit centered heat ball $E(1) = \big\{(x,t)\in \mathbb{R}^{n+1}; ~ t\leq 0, ~ \Phi(x, -t)\geq \frac{1}{r^n} \big\}.$ Compute the following integral: $\int_{E(1)} \frac{|x|^2}{t^2}~\mbox{d}x\mbox{d}t$. (The value should be $4$).
3. Section 2.5: problems 8, 11, 12.
Homework 5 (due March 14)
1. Assume that $u\in\mathcal{C}^m(\mathbb{R}^n\times [0,\infty))$ solve: $u_{tt} = \Delta u$. For a fixed $x$, define a function of two variables $r,t>0$ by the formula: $U(r,t) = \frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)} u(y,t)~\mbox{d}\sigma(y)$.
(i) Prove that $U_{rr}(r,t)= \frac{1}{|\partial B(x, r)|}\int_{\partial B(x,r)} \Delta u(y,t)~\mbox{d}\sigma(y) + \frac{n-1}{n} \frac{1}{|B(x,r)|}\int_{B(x,r)} \Delta u(y,t)~\mbox{d}y$.
(ii) Prove that $\displaystyle{\lim_{(r,t) \to (0, t_0)} U_{rr}(r,t) = \frac{1}{n}\Delta u(x,t)}$.
2. Prove Lemma 2 in Section 2.4.
3. Section 2.4: problems 13, 14, 15.
Homework 6 (due April 6)
1. Prove with all details Theorems 2 and 3 in Section 2.4 (solution of wave equation in odd and even dimensions). Also, prove that when $n$ is an odd integer, then: $ 2\frac{\omega_n}{\omega_{n+1}} = \frac{1\cdot 3\cdot\ldots \cdot (n+1)}{2\cdot 4\cdot\ldots \cdot n}$.
2. Let $H:\mathbb{R}^n\to\mathbb{R}$ be a convex function.
(i) Prove that $H$ must be continuous.
(ii) If additionally $H\in\mathcal{C}^2$, prove that at each $x$ the quadratic form $\nabla^2 H(x)$ is nonnegative definite.
(iii) Prove that for every $y$ there exists $s$ such that: $H(x) \geq H(y) + \langle s, x-y\rangle$ for all $x$.
3. Assume that $H:\mathbb{R}^n\to\mathbb{R}$ is superlinear and uniformly strictly convex (i.e. $\langle \nabla^2H(p) \xi, \xi\rangle\geq \theta |\xi|^2$ for all $p,\xi\in\mathbb{R}^n$ and some $\theta>0$). Prove that the Lagrangian $L=H^*$ satisfies: $\frac{1}{2} L(q_1) + \frac{1}{2} L(q_2) \leq L(\frac{q_1+q_2}{2}) + \frac{1}{8\theta} |q_1 - q_2|^2$ for all $q_1, q_2\in \mathbb{R}^n$.
4. Section 3.5: problems 5 and 6.
Homework 7 (due April 15)
Section 3.5: problems 7, 8, 10. Section 10.4: problems 1, 2.
Homework 8 (due April 27)
1. Section 3.3: Prove, with all details, formulas (43) and (46) in the textbook.
2. Section 10.4: problems 4, 5.
Supplementary reading
"Exploring the Unknown: The Work of Louis Nirenberg on Partial Differential Equations"
by Tristan Riviere.
Longer unedited version.
"Hyperbolic Conservation Laws: An Illustrated Tutorial"
by Alberto Bressan.
The last Fields medalists work.
Check also the slightly longer
Laudatios
at ICM