### 澳洲代写｜MATH6406｜Partial Differential Equations, Fourier Analysis and Complex Analysis偏微分方程、傅立叶分析和复分析 澳洲国立大学

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Many physical processes such as vibrating strings, diffusion of heat and fluid flows are well modelled by partial differential equations and/or integral equations. This course provides an introduction to methods for solving and analysing standard partial differential equations and integral equations, including an introduction to complex analytic techniques.

## Partial Differential Equations偏微分方程问题集

Let $\Omega$ be bounded and assume $c \leq 0$ in $\Omega$. Let $L u \geq 0$ (or, respectively, $L u \leq 0$ ). Then
$$\max {\bar{\Omega}} u \leq \max {\partial \Omega} u^{+} \text {(or, resp., } \min {\bar{\Omega}} u \geq \min {\partial \Omega} u^{-} \text {). }$$
Here, $u^{+}=\max (u, 0), u^{-}=\min (u, 0)$. In particular, if $L u=0$ in $\Omega$, then
$$\max {\bar{\Omega}}|u|=\max {\partial \Omega}|u| \text {. }$$
Proof. If $u \leq 0$ throughout $\Omega$, the corollary is trivially true. Hence we may assume that $\Omega^{+}=\Omega \cap{u>0} \neq \emptyset$. On $\Omega^{+}$, we have $-c u \geq 0$, and hence
$$a_{i j} \frac{\partial^2 u}{\partial x_i \partial x_j}+b_i \frac{\partial u}{\partial x_i} \geq 0 .$$
Hence the previous theorem implies that the maximum of $u$ on the closure of $\Omega^{+}$is equal to its maximum on $\partial \Omega^{+}$. Since $u=0$ on $\partial \Omega^{+} \cap \Omega$, this maximum must be achieved on $\partial \Omega$.

The following corollary is typically used in applications. It yields a uniqueness result as well as a comparison principle.

Assume $L u \geq 0$ ( $L u \leq 0$ ) in $\Omega$ (not necessarily bounded) and assume that $u$ is not constant. If $c=0$, then $u$ does not achieve its maximum (minimum) in the interior of $\Omega$. If $c \leq 0, u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c, u$ cannot be zero at an interior maximum (minimum).
Proof. Assume that $u$ assumes its maximum $M$ at an interior point and let $\Omega^{-}=\Omega \cap{u<M}$. If $\Omega^{-}$is not empty, then $\partial \Omega^{-} \cap \Omega$ is not empty. Let $\mathbf{y}$ be a point in $\Omega^{-}$that is closer to $\partial \Omega^{-}$than to $\partial \Omega$ and let $B$ be the largest ball contained in $\Omega^{-}$and centered at $\mathbf{y}$. Let $\mathbf{x}_0$ be a point on $\partial B \cap \partial \Omega^{-}$. Then we can apply the previous lemma to $B$. We conclude that $\nabla u$ is nonzero at $\mathbf{x}_0$, contradicting the assumption that $u$ assumes its maximum there.

Let $u$ be a solution of $\Delta u=u^3-u$ on a bounded domain $\Omega$. Assume that $u=0$ on $\partial \Omega$. Show that $u \in[-1,1]$ throughout $\Omega$. Can the values \pm 1 be achieved?
Assume that the $n \times n$ matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive semidefinite. Show that $\operatorname{tr}(\mathbf{A B}) \geq 0$. Hint: $\mathbf{B}=\sum_k \lambda_k \mathbf{q}_k \mathbf{q}_k^T$, where the $\lambda_k$ and $\mathbf{q}_k$ are the eigenvalues and eigenvectors of $\mathbf{B}$.

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