Problem 1. Set $\Delta u(x)=1, x \in \overline{B_{2}^{2}(0)} \backslash B_{1}^{2}(0) .$ Then $\int_{S_{1}^{2}(0)} \frac{\partial u}{\partial \rho}(\rho, \theta) ds \quad$ and $\quad \int_{S_{2 }^{2}(0)} \frac{\partial u}{\partial \rho}(\rho, \theta)$Who is bigger?
Proof . 解 注意到
$$\frac{\partial u}{\partial \nu}=\frac{\partial u}{\partial \rho} \quad \text { 当 } s \in S_{2}^{2}(0), \quad \frac{\partial u}{\partial \nu}=-\frac{\partial u}{\partial \rho} \quad \text { 当 } s \in S_{1}^{2}(0)$$

$$3 \pi=\int_{\overline{B_{2}^{2}(0)}} \int_{B_{1}^{2}(0)} \quad 1 d x d y=\int_{B_{2}^{2}(0)} \int_{S_{2}^{2}(0)} \frac{\partial u}{\partial \rho} d s-\int_{S_{1}^{2}(0)} \frac{\partial u}{\partial \rho} d s$$

$$\int_{S_{2}^{2}(0)} \frac{\partial u}{\partial \rho} d s=\int_{S_{1}^{2}(0)} \frac{\partial u}{\partial \rho} d s+3 \pi>\int_{S_{1}^{2}(0)} \frac{\partial u}{\partial \rho} d s$$

Problem 2.

Let

$$\bar{\Omega}=\left\{(x, y) \in \mathbb{R}^{2} \mid 1 \leqslant x^{2}+2 y^{2} \leqslant 2\ right\}; u \in C^{2}(\bar{\Omega})$$

think abut $$\begin{array}{ll} \Delta u(x, y)=0, & (x, y) \in \bar{\Omega}; \\ u(x, y)=x+y, & x^{2}+2 y^{2}=2; \\ \frac{\partial u(x, y)}{\partial \nu}+(1-x) u(x, y)=0, & x^{2}+2 y^{2}=1 \\ \text {Find} \max _{\bar{\Omega}}|u(x, y)|. \end{array}$$

Proof . $\quad$ 根据极值原理, $\max _{\bar{\Omega}}|u(x, y)|$ 在区域的边界上达到. 因此, 必须比较解在边 界上的值.

$0\left(\frac{\partial u}{\partial \nu}\left(\xi_{\min }\right) \leqslant 0\right) .$ 考虑到
$$(1-x) \geqslant 0, \quad \text { 当 } x^{2}+2 y^{2}=1$$

$$\max _{x^{2}+2 y^{2}=2}(x+y)$$

Problem 3.

Let $\Omega=\left\{(x, y) \in \mathbb{R}^{2} \mid 0<x<1,0<y<1\right\}, u \in C^{2 }(\bar{\Omega}),$ $\Delta u=0 \quad$ In $\bar{\Omega}$, $\left.\quad u\right|_{y=0}=\left.u\right|_{y=1} =0 \quad$ when $0 \leqslant x \leqslant 1$ Function $f(x):=\int_{0}^{1} u^{2}(x, y) d y$ Is there an inflection point in the open interval (0,1)?

Proof . 函数 $u^{2}(x, y) \in C^{2}(\Omega),$ 所以 $\int_{0}^{1} u^{2}(x, y) d y$ 可对 $x$ 求两次导数, 那么利用函数 $u$ 的调和性质, 有
$$f^{\prime \prime}(x)=2 \int_{0}^{1}\left(u_{x}^{2}+u u_{x x}\right) d y=2 \int_{0}^{1}\left(u_{x}^{2}-u u_{y y}\right) d y$$

$$f^{\prime \prime}(x)=2 \int_{0}^{1}\left(u_{x}^{2}+u_{y}^{2}\right) d y \geqslant 0, \quad x \in[0,1]$$

Problem 4.

Find all in $\mathbb{R}^{2}$ such that $$u_{x}(x, y)<u_{y}(x, y), \quad \forall(x, y) \in \mathbb{R}^{2}$$ The harmonic function of $u(x, y) .$

Proof .

$$d x=-d y=\frac{d u}{C} \text { . }$$

$$x+y=C_{1}, \quad u-C x=C_{2}$$

$$0=\varphi_{x x}+\varphi_{y y}=2 \varphi^{\prime \prime}$$