Math Problems Vault

Problem 6.1.1

(a)(b)

?

(a)

Let $A = (a_{ij})_{i,j=1}^{n}$ be a real matrix. Show that

\int _{\left|x\right|<1} (x, Ax)\, dx = \frac{\omega _n}{n(n+2)} \operatorname {trace} A.

Here $(\cdot , \cdot )$ is the dot product of vectors in $\mathbb {R}^{n}$ and $\omega_n$ is the area of the unit sphere in $\mathbb {R}^{n}$ .

(b)

Show that $u \in C_0^2(\mathbb {R}^{n})$ implies

\int _{\mathbb {R}^{n}} (\Delta u)^2\, dx = \sum _{i,j=1}^{n} \int _{\mathbb {R}^{n}} \left|D_{ij} u\right|^2\, dx.

Problem 6.1.2

Let $T$ be a distribution on $\mathbb {R}$ and suppose that $T' = 0$ on $\mathbb {R}$ . Show that $T = \operatorname {const}$ ; i.e., show that there is a number $a$ such that

T(\phi ) = \int _{\mathbb {R}} a \phi \, dx \quad \text{for all } \phi \in C_0^{\infty }(\mathbb {R}).

?

Problem 6.1.3

Let $u \in H^s(\mathbb {R}^{n})$ with $s > n/2$ . Show that $\lim_{\left|x\right| \to \infty } u(x) = 0$ .

?

Problem 6.1.4

Let $u$ be defined for $f \in \mathcal{D}((0,1))$ by

\langle u, f \rangle = \sum _{n=0}^{\infty } f^{(n)}\left(\frac{1}{n}\right).

Determine whether $u$ is a distribution on $(0,1)$ , and support your answer.

?

Problem 6.1.5

Let

B = \left\{ (x,y) \mid x^2 + y^2 < 1\right\} .

For which $p \geq 1$ does the function

u(x,y) = \frac{x}{\sqrt{x^2+y^2}}

belong to $W^{1,p}(B)$ ?

?

Problem 6.1.6

Let $\Sigma$ be a smooth hypersurface dividing $\mathbb {R}^{n}$ into two disjoint open regions $\Omega_1$ and $\Omega_2$ . Denote by $\nu = \nu (x)$ the unit normal to $\Sigma$ pointing into $\Omega_2$ . Suppose $v : \mathbb {R}^{n} \to \mathbb {R}^{n}$ solves the equation

\operatorname {div} v(x) = b(x) \quad \text{for } x \in \mathbb {R}^{n} - \Sigma \tag {*}

in the classical ( $C^1$ ) sense where $b$ is a continuous function. Assume furthermore that $v$ lies in both $C^0(\overline{\Omega_1})$ and $C^0(\overline{\Omega_2})$ (but not necessarily $C^0(\mathbb {R}^{n})$ !!) with

v^{+}(x_0) \equiv \lim _{\substack {x \to x_0 \\ x \in \Omega _1}} v(x), \qquad v^{-}(x_0) \equiv \lim _{\substack {x \to x_0 \\ x \in \Omega _2}} v(x)

for each $x_0 \in \Sigma$ . Derive a necessary and sufficient condition in terms of $v^+$ , $v^-$ and $\nu$ for $v$ to be a distribution solution of (*) on all of $\mathbb {R}^{n}$ .

?

Problem 6.1.7

(a)(b)

Recall the definition of the curl operator in two dimensions acting on a vector field $U(x,y) = (u(x,y), v(x,y))$ :

\nabla \times U \equiv u_y - v_x.

?

(a)

Give a definition of what it means for a (not necessarily continuous) vector field $U$ to have curl zero in the sense of distributions.

(b)

Suppose $U$ takes the form

U(x,y) \equiv \begin{cases} U_1, & (x,y) \in \Omega _1, \\ U_2, & (x,y) \in \Omega _2, \end{cases}

where $\Omega_1$ and $\Omega_2$ are open sets such that $\overline{\Omega_1} \cup \overline{\Omega_2} = \mathbb {R}^{2}$ , $\Gamma \equiv \partial \Omega_1 \cap \partial \Omega_2$ is a smooth curve, and $U_1$ and $U_2$ are (different) constant vectors. If $U$ has curl zero in $\mathbb {R}^{2}$ in the sense of distributions, describe as completely as possible the curve $\Gamma$ .

Problem 6.1.8

Let $p$ be a polynomial in $n$ variables, and assume that the set $\left\{ x : p(ix) = 0\right\}$ is bounded. Prove that there exists a tempered distribution $E$ on $\mathbb {R}^{n}$ such that $p(D)E - \delta \in S$ . Here $S$ is the Schwartz class of rapidly decreasing functions, and $\delta$ is the Dirac distribution defined by $\delta \cdot \phi = \phi (0)$ .

?

Problem 6.1.9

Let $P(D)$ be an $m$ -th order linear constant coefficient partial differential operator on $\mathbb {R}^{n}$ ; i.e.,

P(D) = \sum _{\left|\alpha \right| \leq m} C_{\alpha } D^{\alpha }

where $\alpha$ denotes a multi-index $\alpha = (\alpha_1, \alpha_2, \ldots , \alpha_n)$ and the $C_{\alpha }$ are constants. Suppose

P(i\xi ) \equiv \sum _{\left|\alpha \right| \leq m} C_{\alpha } (i\xi )^{\alpha } \neq 0 \quad \text{for all } \xi \in \mathbb {R}^{n} - \left\{ 0\right\} .

If $u \in S'$ satisfies $P(D)u = 0$ , prove that $u$ must be a polynomial. (Here $S'$ denotes the space of tempered distributions on $\mathbb {R}^{n}$ .)

?

Problem 6.1.10

(a)(b)(c)(d)

?

(a)

Show that

\lim _{t \to 0^+} \frac{1}{\sqrt{4\pi t}} e^{-x^2/4t} = \delta (x)

in $\mathcal{D}'$ .

(b)

Show how to define $\frac{1}{x}$ as a distribution (i.e., define the Cauchy Principle Value and show that it is in $\mathcal{D}'$ ).

(c)

Calculate rigorously the derivative of $\log \left|x\right|$ and express the result in terms of your answer to (b).

(d)

Calculate the Fourier transform of the distribution $\frac{1}{x}$ .

Problem 6.1.11

(a)(b)(c)

?

(a)

Let

Lu(x) = \sum _{\left|\alpha \right| \leq m} a_{\alpha } D^{\alpha } u(x), \quad x \in \mathbb {R}^{n}

where the $a_{\alpha }$ 's are constants, and $\alpha$ is a multi-index. What is a fundamental solution of $L$ ? When is it unique?

(b)

If $I_+$ denotes the Heaviside step function,

I_+(\xi ) = \begin{cases} 1, & \xi \geq 0, \\ 0, & \xi < 0, \end{cases}

show that

F(x,t) = \frac{1}{2} I_+(t) I_+(t - \left|x\right|)

is a fundamental solution of the wave operator $W(u) = u_{tt} - u_{xx}$ on $\mathbb {R}^{2}$ .

(c)

Express the solution of the Cauchy problem

u_tt - u_xx = (x,t), -< x < , t > 0 u(x,0) = f(x), u_t(x,0) = g(x), -< x <

( $f, g, \phi$ sufficiently nice) in terms of the above fundamental solution. Simplify your result to obtain D'Alembert's solution of this problem.

Problem 6.1.12

Let $u$ be the solution of the initial problem

u_tt = -u_x_j x_j x_k x_k, x R^3, t > 0, u(x,0) = 0, u_t(x,0) = (x),

where summation is understood in the pde, and $\phi$ is a $C^{\infty }$ function on $\mathbb {R}^{3}$ with compact support. Assume that there is a constant $C$ such that

\left\| u(\cdot ,t)\right\| _{L^2(\mathbb {R}^{3})} \leq C t^{1/4} / \log t,

and prove that

\int \phi (x)\, dx = 0.

?

Problem 6.1.13

Given $\phi \in S$ (rapidly decreasing functions) over $\mathbb {R}^{1}$ , consider the solution $u$ of the Schrodinger equation

iu_t + u_xx = 0 (x R^1, t > 0), u(x,0) = (x).

Show that

\lim _{t \to \infty } \int _{\left|x\right|<t} \left|u(x,t)\right|^2\, dx = \int _{\left|\xi \right|<\frac{1}{2}} \left|\hat{\phi }(\xi )\right|^2\, d\xi .

Here $\hat{\phi }$ denotes the Fourier Transform of $\phi$ .

?

Problem 6.1.14

Suppose $u$ is a smooth solution of the following problem

u_xxt + u_xx - u^3 = 0, in [0,1] [0,), u(0,t) = u(1,t) = 0 for t [0,)

with initial data $u(x,0) = x(x-1)$ . Derive a differential inequality for $w := \int_0^1 \left|u_x(x,t)\right|^2\, dx$ , and show that $u(x,t)$ uniformly tends to zero as $t \to \infty$ .

?

Problem 6.1.15

(a)(b)

Assume that $g \in L^2(\mathbb {R})$ and consider the Cauchy problem

u_t + u_x = \varepsilon u_{xxx}, \quad u(0,x) = g(x). \tag {$CP$)$_{\varepsilon }$}

?

(a)

For every $\varepsilon$ , write a formula for the solution $u^{\varepsilon }(t,x)$ using Fourier transform. Show that

\left\| u^{\varepsilon }(t,\cdot )\right\| _{L^2(\mathbb {R})} = \left\| g\right\| _{L^2(\mathbb {R})} \quad \forall t \geq 0.

(b)

Prove that, as $\varepsilon \to 0$ , the solutions $u^{\varepsilon }$ of $(CP)_{\varepsilon }$ converge to the function $v(t,x) = g(x-t)$ . More precisely, show that $\left\| u^{\varepsilon }(t) - v(t)\right\|_{L^2(\mathbb {R})} \to 0$ , for every fixed time $t$ .