## 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.