MATH 236 - Partial Differential Equations

January 16

Topics

1. Introduction to the course.
2. The Spaces \(C^r(\Omega)\), multivariate Taylor series

January 18

Topics

1. Normed Spaces
2. Function Spaces

Readings

1. Ch. 1 of Hunter's notes (today I covered only normed spaces, next lecture I will go over metric ones)
2. Function Spaces, by Terry Tao (UCLA)
3. An interesting discussion on why the natural topology on \(C^\infty([a,b])\) cannot come from a norm
4. An interesting discussion on the infinite dimensional vector space \(\mathbb R_\Bbb Q\).
5. If you are curious enough, read the following notes on Hamel bases of vector spaces by K. Smith (UMich).

Homework #1, due Friday, Feb 1

1. Consider the sequence of continuous piecewise linear functions in $C^0([0,1])$ defined by $$ f_n(x) = \begin{cases}0,&x\in[0,\frac{1}{2}-\frac{1}{n}]\\ a_nx+b_n,&x\in[\frac{1}{2}-\frac{1}{n},\frac{1}{2}+\frac{1}{n}]\\ 1,&x\in[\frac{1}{2}+\frac{1}{n},1]\\\end{cases} $$ where the coefficients $a_n$ and $b_n$ are chosen so that $f_n$ is continuous.
   (a) Find the values of $a_n$ and $b_n$;
   (b) Show that this is not a Cauchy sequence with respect to the $L^\infty$ norm $$ \|f\|_{L^\infty} = \sup_{x\in[0,1]}|f(x)|\,; $$ (c) Show that, on the contrary, this is a Cauchy sequence with respect to the $L^p$ norm $$ \|f\|_{L^p} = \left[\int\limits_0^1|f(x)|^p\,dx\right]^{1/p} $$ for any $1\leq p<\infty$.
   (d) Finally show that, considering $C^0([0,1])$ as a subset of the normed vector space $(L^p([0,1]),\|\cdot\|_{L^p})$ of all integrable functions with finite $L^p$ norm, the sequence $\{f_n\}$ converges to the (integrable but discontinuous) function $$ f(x) = \begin{cases}0,&x\in[0,\frac{1}{2})\\ 1,&x\in(\frac{1}{2},1]\\\end{cases} $$ (the value of $f$ at $x=1/2$ is irrelevant since functions in $L^p$ spaces are considered the same whenever they differ in a set of measure zero).
2. Prove that $(C^0([0,1]),\|\cdot\|_{L^\infty})$ is a complete normed vector space (in short, a Banach space), namely prove that every uniformly Cauchy sequence $\{f_n\}\subset C^0([0,1])$ converges to a continuous function over $[0,1]$.
3. Prove that $(C^1([0,1]),\|\cdot\|_{L^\infty})$ is not complete by building a Cauchy sequence $\{f_n\}\subset C^1([0,1])$ that does not converge to a $C^1([0,1])$ function in the $L^\infty$ norm (no need to write analytical formulas, a good picture would be enough).
   In fact, by the Stone-Weierstrass theorem, we know that $$ \overline{C^r([0,1])}^{\|\cdot\|_{L^\infty}}=C^0([0,1]) $$ for any $r=0,1,2,\dots,\infty,\omega$ since every continuous function is the uniform limit of a sequence of polynomials.
4. Prove that, on the contrary, $(C^1([0,1]),\|\cdot\|_{C^1})$, with $$ \|f\|_{C^1} = \sup_{x\in[0,1]}|f(x)| + \sup_{x\in[0,1]}|f'(x)|\,, $$ is a Banach space, namely show that, given any uniformly Cauchy sequence $\{f_n\}$ of $C^1$ functions such that also the sequence of their derivatives $\{f'_n\}$ is uniformly Cauchy in $[0,1]$, $\{f_n\}$ does converge to a $C^1$ function over $[0,1]$.

January 23

Topics

1. The Banach spaces $(C^k([0,1],\|\cdot\|_{C^k}), k=0,1,2,\dots$
2. Metric spaces.
3. Natural topologies on $C^\infty([0,1])$ and $C^k(\Bbb R)$.

Readings

1. Metric Spaces by J.Hunter (UCDavis)
2. The Banach spaces $C^k([a,b])$ by P. Garrett (UMN).

January 25

Topics

1. Again about the $C^k$ spaces.
2. Discussion on the completeness of $C^1(K)$.

Readings

1. See Thm 5.18 in Sequences and series of functions by J.Hunter (UCDavis)

Homework

1. Prove that $(C^2([0,1]),\|\cdot\|_{C^2})$ is a Banach space.

January 28

Topics

1. Definition of the spaces $W^{k,p}$ and $L^p_{loc}$.
2. Continuity of the differential operators.

Readings

1. My (very short) notes on Function Spaces
2. Evans, Appendix C

January 30

Topics

1. Definition of the spaces $W^{k,p}$ and $L^p_{loc}$.
2. Continuity of the differential operators.
3. Mollifiers

Readings

1. My (very short) notes on Function Spaces
2. Evans, Appendix C

February 1

Topics

1. Definition of the spaces $W^{k,p}$ and $L^p_{loc}$.
2. Continuity of the differential operators.

Readings

1. My (very short) notes on Function Spaces
2. Evans, Appendix C

February 4

Topics

1. Definition of a general PDE of order $k$.
2. Digression: The dual of a finite-dimensional linear vector space.

Readings

1. Evans, Chapter 1.
2. Vector and coVectors on math.stackexchange.
3. Covariance and contravariance of vectors on wikipedia.

February 6

Topics

1. Solving the linear 1st order PDE $$ X^\alpha(x)\partial_\alpha u(x) + \mu(x) u(x)= v(x)\,. $$ The case $X=\partial_x$ and $\Omega=\Bbb R^2$.

Readings

1. My (very short) notes on PDEs
2. Linear 1st order PDEs on NPTEL (India)

February 8

Topics

1. Digression: action on vectors and covectors induced by a map $f:\Bbb R^n\to\Bbb R^m$; $\{\partial_\alpha|_{x=x_0}\}$ and $\{dx^\alpha\}$ as bases of, respectively, the spaces of tangent and cotangent vectors at $x_0$.
2. Influence of the topology of the integral trajectories of a vector field $X(x)$ on the regularity of the solutions of the linear 1st order PDE $$ X^\alpha(x)\partial_\alpha u(x) = v(x)\,. $$ The case $X=\partial_x$ and $\Omega=\Bbb R^2\setminus\{0\}\times[0,\infty)$.

Readings

1. My (very short) notes on PDEs
2. Here are some notes on flows of vector fields by J. Marsden (CalTech).

Homework #2, due Wednesday, Feb 20

1. Solve the PDE $$ -yu_x+xu_y = 0,\;u(x,x^2)=x^3\;\;\hbox{ for }\;\;x>0, $$ in $\Omega=\Bbb R^2\setminus\{(0,0)\}$.
2. Solve the PDE $$ 2 u_x+3 u_y + 8u = 0,\;u(x,0)=\sin x $$ in $\Omega=\Bbb R^2$.
3. Solve the PDE $$ xu_x+yu_y = \sqrt{x^2+y^2},\; u=\frac{1}{2}\;\hbox{ on the circle }\;\{x^2+y^2=1\} $$ in $\Omega=\Bbb R^2\setminus\{(0,0)\}$.
4. Show that the PDE $$ yu_x-xu_y = 1,\;u(0,y)=0 $$ in $\Omega=\Bbb R^2\setminus\{(0,0)\}$ does not admit a $C^0$ solution (Hint: assume such $u$ exists and study how its value changes along the integral trajectories of $X$).
5. Show that the PDE above does admit, on the contrary, a $C^\infty$ solution in $\Omega=\Bbb R\times(0,\infty)$.

February 11 - March 8

Topics

1. Quasi-linear 1st order PDEs
2. Linear and non-linear wave equations
3. Fully nonlinear 1st order PDEs
4. A very short introduction to Lagrangian and Hamiltonian mechanics and the least action principle
5. The Hamilton-Jacobi equation

Readings

1. My short notes on PDEs
2. The whole Chapter 3 of Evans. His notation is different from the one I use in the notes but it's healthy for you all to see different ones. We did not cover (yet) all topics in that chapter, here is a list of sections you can skip (although I suggest you to give them a look at a second read): 3.1.2 (envelopes), 3.3.2 (Legendre transformation, Hopf-Lax formula), 3.3.3 (weak solutions), 3.4 (Conservation laws)
3. Schrodinger's train of thoughts, a very nice and readable article on the interplay between classical and quantum mechanics and the fundamental role in it of the Hamilton-Jacobi equation and the action functional.

Take-home Midterm, due Monday, Mar 18

1. Solve the quasilinear PDE $$ u_x+uu_y = 2x,\;u(0,y)=y^2 $$ in $\Omega=\Bbb R^2$.
2. Solve the fully nonlinear PDE $$ u^{\color{red}{2}}_x+u_y = u,\;u(x,0)=1+x^2 $$ in $\Omega=\Bbb R^2$.
3. Solve the Hamilton-Jacobi PDE $$ u_xu_y = xy,\;u(x,0)=\color{red}{x} $$ in $\Omega=\Bbb R^2$.
4. Let $u$ be the solution of the Burgers quaslilinear 1st order PDE $$ u_t+uu_x=0,\;u(0,x)=g(x)\in C^2(\Bbb R^2) $$ and suppose that $\|g\|_{C^1(\Bbb R)} < \infty$ and that $u\in C^2((-T,T)\times\Bbb R)$. Prove that:
   1. $g\geq0\implies u\geq0$ ;
   2. if $g$ has compact support, then so does $u(t,\cdot)$ for all $t\in(-T,T)$;
   3. if $g\geq0$ and has compact support, then $$ \|u(t,\cdot)\|_{L^p} = \|g\|_{L^p(\Bbb R)}\,,\;\;\forall t\in(-T,T) $$
   4. if $g\geq0$ and has compact support, then $$ \int_\Bbb R f(u(t,x))dx = \int_\Bbb R f(g(x))dx $$ for all $f\in C^1([0,\infty))$.
5. Let $u$ be the solution of the Burgers quaslilinear 1st order PDE $$ u_t+uu_x=0,\;u(0,x)=g(x)\in C^2(\Bbb R^2) $$ and suppose that $g'\geq0$ and has compact support (i.e. $g$ is constant outside some compact set). Prove that:
   1. the local solution $u(t,x)$, $t\in(-T,T)$, can be extended to all $t\geq0$;
   2. $u_x(t,x)\geq0$, and $u_x(t,\cdot)$ has compact support for all $t\geq0$;
   3. the function $f(t)=\|u(t,\cdot)\|_{L^2(\Bbb R)}$ is decreasing.
6. I often mentioned in class that we can always assume locally that every smooth hypersurface $\Gamma\subset\Bbb R^n$ can be written locally as $x^n=0$. Prove this fact on the plane, namely let $\Gamma\subset\Bbb R^2$ be the regular level set of a function $h(x,y)$ (namely $dh\neq0$ on every point of $\Gamma$) and show that, close to any point of $\Gamma$, we can find coordinates $(x',y')$ where $\Gamma$ writes as $y'=0$.
   Hint: you will need to use the implicit function theorem.
7. Now that you have some experience with 1st order PDEs, go on and prove the following local existence and uniqueness theorem in two variables:
   Let $F\in C^3(J^1(\Bbb R^2,\Bbb R))$, $\Gamma_0=\Bbb R\times\{0\}$, $g\in C^3(\Bbb R)$ and $(x_0,0,g(x_0),g'(x_0),p_{y,0})$ a point of $F_0=\{F=0\}$ where $\partial_{p_y}F\neq0$. Then there exists a neighborhood $U\subset\Bbb R^2$ of $(x_0,0)$ and a unique function $u\in C^2(U)$ such that:
   1. $F(x,y,u(x,y),u_x(x,y),u_y(x,y))=0\hbox{ for all }(x,y)\in U$;
   2. $u(x,0)=g(x)\hbox{ for all }(x,0)\in U$;
   3. $u_y(x_0,0)=p_{y,0}$.
   
   Hint: Existence. The only thing we did not cover is showing that the surface $\hat\Gamma=\cup_{t}\hat\Gamma_t$ in the notes is really the 1-graph of a function $u(x,y)$, namely you need to show that $p_\alpha(t)=\partial_x u(x(t),y(t))$ for all $t$. Uniqueness. Assume there is a second $C^2$ solution $v$ and check what happens to the restrictions of $u$ and $v$ along the projection $(x(t),y(t))$ of the integral trajectories of $X_F$. Essentially, the uniqueness for this PDE comes from the uniqueness of ODEs solutions.

March 18

Topics

1. A special case of quasilinear 1st order PDEs: Conservation Laws.
   • Weak solutions
   • Rankine-Hugoniot Condition

Readings

1. notes on conservation laws by S. Baskar
2. notes on PDEs by Y. Gorb
3. Evans, pp. 135-139

March 20

Topics

A special case of quasilinear 1st order PDEs: Conservation Laws.
• Shock waves.

Readings

1. notes on conservation laws by S. Baskar
2. notes on PDEs by Y. Gorb
3. Evans, pp. 139-143

March 22

Topics

A special case of quasilinear 1st order PDEs: Conservation Laws.
• The Lax-Oleinik formula

Readings

1. notes on conservation laws by S. Baskar
2. notes on PDEs by Y. Gorb
3. Evans, pp. 143-148

March 25

Topics

1. General 2nd order PDEs and the Cauchy-Kovaleskaya theorem.

Readings

1. notes on PDEs by L. Erdos
2. Evans, Section 4.6

March 27

Topics

1. General 2nd order PDEs and the Cauchy-Kovaleskaya theorem.

Readings

1. notes on PDEs by L. Erdos
2. Evans, Section 4.6

March 29

Topics

1. Classification of PDEs

Readings

1. notes on PDEs by L. Erdos
2. The Cauchy-Kovalevskaya Theorem by D. Gaidashev
3. On the local solvability of PDEs with simple characteristics by Yu. Egorov

April 1-5

Topics

1. More about the Cauchy-Kovalevskaya theorem

Readings

1. The Cauchy-Kovalevskaya Theorem by D. Gaidashev

April 8-12

Topics

• Laplace and Poisson equations:
  1. Fundamental solution of the Laplace equation
  2. Solution of the Poisson equation through convolution
  3. Mean-value formulas
  4. Maximum principle
  5. Uniqueness of the solution
  6. Smoothness of the solution
  7. Estimates on the derivatives
  8. Liouville's theorem
  9. Analiticity of harmoni functions
  10. Harnack's inequality

Readings

1. Evans textbook, Section 2.2

April 15-19

Topics

• Heat equation:
  1. Fundamental solution of the homogeneous problem
  2. Solution of the non-homogeneous problem (Duhamel principle)
  3. Mean-value formula
  4. Maximum principle
  5. Uniqueness of the solution
  6. Regularity of the solution

Readings

1. Evans textbook, Section 2.3

April 22-26

Topics

• Wave Equation:
  1. D'Alembert formula
  2. Domain of dependence and Domain of influence
  3. Solving the Poisson equation in 1+1 dimensions via coordinate change
  4. Solving the Poisson equation in 1+n dimensions via Duhamel's principle

Readings

1. notes on Hyperblic PDEs by J. Shatah
2. Evans textbook, Section 2.4

Take-home Final, due Tuesday, April 30

1. Let $v\in C^2(\Omega)\cap C^0(\bar\Omega)$ be a subharmonic function, namely a solution of the partial differential inequality $$\Delta v\geq0.$$ Modify the proof in Evans for harmonic functions to prove that $$ v(x) \leq \frac{1}{Vol(B_r(x))}\int_{B_r(x)}v(y)dy $$ for all $x\in\Omega$ and $r>0$ such that the ball $B_r(x)$ is entirely contained in the open set $\Omega\subset\mathbb R^n$. Then show that, even for subharmonis functions, $$ \max_{x\in\bar\Omega}v(x) = \max_{x\in\partial\Omega}v(x). $$
2. Prove the non-uniqueness of solutions of the Heat equation in 1+1 dimensions by showing that every series $$ u_k(x,t) = \sum_0^\infty \frac{g_k^{(n)}(t)}{(2n)!}x^{2n}, $$ with $$ g_k(t) = \begin{cases}e^{-\frac{1}{t^k}},&t>0\\ 0,&t=0\end{cases},\;k>1, $$ converges and solves the boundary value problem $$ u_t=u_{xx},\;u(0,x) = 0. $$ Help: verifying that the series formally solves the heat equation is easy. In order to prove the convergence of the series, use the fact that $$ |g_k^{(n)}(t)|\leq\frac{n!}{(\theta t)^n}e^{-\frac{1}{t^k}} $$ for some $\theta$ depending only on $k$ to find an upper bound for the series $$ \sum_0^\infty \left|\frac{g_k^{(n)}(t)}{(2n)!}x^{2n}\right|. $$
3. Consider the 1D wave equation $$ u_{tt}-u_{xx}=0,\;u(0,x)=g(x),\;u_t(,x)=h(x), $$ where $g$ and $h$ have compact support. Prove that the function $$ E(t) = \int_{\mathbb R}\left[u_t^2(t,x) + u^2_x(t,x)\right]dx $$ is constant for any solution of the PDE above. Then prove also that, for $t$ large enough, both terms $$ K(t) = \int_{\mathbb R}u_t^2(t,x) dx, \;\; P(t) = \int_{\mathbb R}u_x^2(t,x) dx $$ are actually constant and equal to each other (equipartition of energy).