Xem các phần khác tại đây.

3 types of PDE

Tại sao lại phân loại, phân loại làm sao?

$$Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y +Fu = G$$

The discriminant

$$d = B^2(x_0,y_0) - 4A(x_0,y_0)C(x_0,y_0)$$

at that point. At $(x_0,y_0)$ the equation is said to be

• Elliptic if $d<0$
• Parabolic if $d=0$
• Hyperbolic if $d>0$

If this is true at all points in a domain $\Omega$, then the equation is said to be elliptic, parabolic, or hyperbolic in that domain.

Distribution sens

[Distribution spaces.pdf] Locally integrable on I $\Leftrightarrow \int_I v\, dx$.

$v=u’$ in the weak sense if

$$-\int_{\Omega} u\varphi' = \int_{\Omega}v \varphi, \forall \varphi \in C_0^{\infty}(\Omega).$$

here $C_0^{\infty}(\Omega)$ denotes the space of $C^{\infty}$ functions on $\Omega$ with compact support in $\Omega$.

If

$$-\int_{\Omega} u \varphi' = T_{\varphi}, \forall \varphi\in C_0^{\infty}(\Omega)$$

then $u’=T$ in distribution sense.

The support of $f$ is the complement of the largest open set where the function is zero.

Ví dụ kinh điển cho thấy sức mạnh của distribution chính là hàm Heaviside function

$$H(x) = \begin{cases} 1 (x>0)\\ 0 (x\le 0) \end{cases}$$

Không tồn tại $v$ để

$$-\int_{\Omega} H\varphi' = \int_{\Omega}v \varphi, \forall \varphi \in C_0^{\infty}(\Omega).$$

Tuy nhiên tồn tại $T_{\varphi}=\varphi(0):=\delta_0$ (theo nghĩa distribution sens)

$$-\int_{\Omega} u \varphi' = T_{\varphi}, \forall \varphi\in C_0^{\infty}(\Omega)$$

$L^p$ and bounded

Câu trả lời này rất hay. Nằm trong không gian $L^p$ chưa chắc là bounded. Ở đây cũng nói về essential bounded luôn.

Lipschitz domain

Xem định nghĩa toán học dài ngoằn của Ronald W. Hoppe ở đây (Definition 2.1).

[wiki] Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. Consequently, many partial differential equations and variational problems are defined on Lipschitz domains.

---

[s] Lipschitz condition, function: Let $\varphi: R^{n-1} \to R$ be a function which satisfies the lipschitz condition

$$\vert \varphi(y') - \varphi(x') \vert \le M \vert y'-x'\vert, \forall y',x' \in R^{n-1}$$

Dual space

Why we need dual space?

Có thể đọc câu trả lời này.

[s] The dual space is always complete (regardless of whether X is complete or not)

Some popular dual space

The dual space of $H^1_0(\Omega)$, denoted $H^{-1}(\Omega):=H^1_0(\Omega)’$ with norm

$$\Vert {f}\Vert_{-1}:=\sup_{v\in H^1_0(\Omega)}\dfrac{f(v)}{ \Vert {v}\Vert_{H^1(\Omega)} }$$

Trong note của Pascal Frey,

$$\begin{align*} \Vert {u}\Vert_{H^{-1}} &= \sup_{\Vert {u}\Vert_{H^0_1}=1} \langle u,v\rangle_{H^{-1},H^1_0} \\ \Vert {u}\Vert_{H^{-1}} &= \sup_{\Vert {u}\Vert_{H^0_1}=1}\int_{\Omega}uv \end{align*}$$

We have the following inclusion rule $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$. $\Leftarrow$ Cái này là Triple Gelfand.

Gelfand triple

[s] Loosely speaking, a rigged Hilbert space (or Gelfand triplel) is a triple $(\Phi, H, \Phi^{\ast})$ where $\Phi$ is a dense subspace of $H$ and $\Phi^{\ast}$ its continuous dual space.

$$\Phi \subseteq H, H^{\ast} \subseteq \Phi^{\ast}.$$

---

I am trying to understand the Hilbert triple $V \subset H \subset V^*$, where $V$ and $H$ are Hilbert spaces and the star denotes the dual space. Eg: $H^1 \subset L^2 \subset H^{-1}.$

Other

ess sup (essential supremum)

$$\mathrm {ess} \sup f=\inf \ \{M:\mu (\{x:f(x)<M\})=0\}$$

ess inf (essential infimum)

$$\mathrm {ess} \inf f=\sup\{m:\mu (\{x:f(x)<m\})=0\}$$

Xem các phần khác tại đây.