Problem set 8

Problem 1¶

Let $T$ be a distribution, show that if $T'=0$, then $T$ is a constant, i.e. for every $\varphi \in \mathscr{C}_c^{\infty}$ $(T,\varphi) = c\int \varphi$ for some constant $c$.

Problem 2¶

In the lecture we showed that $\frac{\partial }{\partial x}[H(x)] = \delta$ directly by definition. In this problem we take a closer look at this fact and get some intuition.

(1) Construct a “smooth version” of the Heviside function, i.e. find a function $h$ such that

(2) Similar to the construction of smooth good kernel, consider the family of functions $h_\epsilon(x) = h(\frac{x}{\epsilon})$. Show that $h_\epsilon \to H$ in $\mathcal D'$, i.e. for every $\varphi \in \mathcal D$, $(T_{h_\epsilon},\varphi)\to (H,\varphi)$.

(3) What can you conclude if you take derivative on (2) ?

(4) Let $\chi_{(-a,a)}$ be the characteristic function of the interval $(-a,a)$, calculate $\frac{d}{dx}[\chi_{(-a,a)}]$.

Problem 3¶

This problem is a high dimensional generalization of $\frac{\partial }{\partial x}[H(x)] = \delta$. In high dimension there are many directions to take derivative and things becomes more complicated but turns out to be more intuitive.

Let $D$ be a bounded domain in $\mathbb{R}^n$. We say that $D$ has $\mathscr{C}^1$ boundary if For every $x_0\in \partial D$, there exists a $\mathscr{C}^1$ function ρ on an open neighbourhood $U$ of $x_0$ such that

• $\rho(x_0) = 0$.
• $d\rho(x_0)\neq 0$.
• $D\cap U = \{\rho < 0\}$.

Here $d$ is the differential $df = \frac{\partial f}{\partial x_1} dx_1 + \cdots + \frac{\partial f}{\partial x_n} df_n$, which can be identified with the gradient of $f$.

(1) Show that if $D$ is a bounded domain with $\mathscr{C}^1$ boundary, then there exists a $\mathscr{C}^1$ function ρ on $\mathbb{R}^n$ such that

• $D = \{\rho < 0\}$.
• $\rho(x) = 0$ and $d\rho(x) \neq 0$ for every $x\in \partial D$.

Let $\chi_D$ be the characteristic function of $D$. We want to calculate $\partial_j[\chi_D]$, where we use $\partial_j$ to denote the partial derivative $\frac{\partial}{\partial x_j}$ for simplicity.

(2) Let $p\in \partial D$ with $\partial_1\rho(p)\neq 0$. Then on a neighbourhood $U$ of $p$, there exists ψ such that $\rho(x) = 0\iff x_1 = \psi(x')$, where $x'$ means $(x_2,\dots,x_n)$. ALso, $\partial_1\rho(x_0) > 0$ (resp. $< 0$) $\implies$ $D$ is given by $x_1<\psi(x')$ (resp. $x_1 > \psi(x')$).

(3) Let $p$ and $U$ be from (2) such that $D$ is given by $x_1>\psi(x')$ on $U$. Draw a picture of $\chi_D, h(x_1 - \psi(x'))$ and $h_\epsilon(x_1 - \psi(x'))$ where $h$ and $h_\epsilon$ are from Problem 2. Show that

in $U$ pointwise and in the sense of distributions.

(4) Let $\varphi \in \mathscr{C}_c^{\infty}(U)$. Show that $(\partial_j\chi_D,\varphi) = \lim_{\epsilon\to 0} \int \nu_j \cdot h_{\epsilon}'(x_1 - \psi(x'))\cdot \varphi(x)dx = \int \nu_j(x')\varphi(\psi(x'),x')dx'$, where $\nu =(1,-\frac{\partial}{\partial x'}\psi) = (1,-\partial_2\psi,\dots,-\partial_n\psi)$. Where $h'_\epsilon(t)$ is the derivative of $h_\epsilon$ given by $\frac{1}{\epsilon}h'(\frac{t}{\epsilon})$.

(5) Recall that the Euclidean surface element $dS$ on $\partial D$ is given by $\sqrt{1+|\psi'|^2}dx'$ on $U$ and $n:= \frac{\nu}{\sqrt{1+|\psi'|^2}}$ is the invards normal on $U\cap \partial D$. So (4) shows that $\partial_j \chi_D = n_j dS$ on $U$ (this means for every $\varphi \in \mathscr{C}_c^{\infty}(U)$,...). Use partition of unity to conclude that

on $\mathbb{R}^n$ (or any neighbourhood of $D$) <- This means for every $\varphi \in \mathscr{C}_c^{\infty}(\mathbb{R}^n)$ or $\mathscr{C}_c^{\infty}(X)$ for open $X\supset D$,...

(6) Prove the Gauss-Green formula

(7) Let $u$ be a $\mathscr{C}^1$ function on a neighbourhood of $D$. Show that $\partial_j[u\chi_D] = (\partial_j u)\chi_D + u n_j dS$.

(8) Prove the integration by parts formula in high dimensional case: Let $\psi\in \mathscr{C}^{1}(\mathbb{R}^n)$, then $\partial_i T_{\psi} = T_{\partial_i\psi}$.

Appendix: Surface measure¶

Case of graph of function¶

Let $z = \psi(x,y)$ for $\mathscr{C}^1$ function ψ and $d\psi\neq 0$ everywhere, let $V$ be a domain on the $x,y$-plane, then the graph of ψ determines a surface patch $S$ in $\mathbb{R}^3$. Let $f$ be a continuous function on $\mathbb{R}^3$, then the surface integral of $f$ on $S$ is given by $\int_{V}f(x,y)\sqrt{1+\psi_x^2+\psi_y^2} \;dxdy$. Similarly, in $\mathbb{R}^n$ and hypersurface determined by graph $x_n = \psi(x_1,\dots,x_n)$ for $(x_1,\dots,x_{n-1})\in V\subset \mathbb{R}^{n-1}$, we let $(dS,f) := \int_S f dS = \int_{V}f(x_1,\dots,x_{n-1},\psi(x_1,\dots,x_{n-1}))\sqrt{1+\partial_1\psi ^2+\cdots+\partial_{n-1}\psi^2}dx_1\cdots dx_{n-1}$.

General case¶

Let $U$ be an open set on which $\partial_1 \rho > 0$, let ψ be the function given in problem 3, so $D$ is given by $x_1<\psi(x')$ on $U$. In other words, we have written $\partial D$ as graph of function on $U$. Following previous discussion, let φ be a continuous function supported on $U$, define $(dS,\varphi)$ to be $\int_{V\subset \mathbb{R}^{n-1}} \varphi(\psi(y),y)\sqrt{1+|\psi'(y)|^2}dy$, where $V$ is the corresponding domain for ψ.

Choose an open cover $(U_j)_{j\in J}$ and use partition of unity $(\varphi)_j$ subordinate to $U_j$ we can define $(dS,f) = \sum_{j\in J}(dS,\varphi_j f)$.