## 澳洲代写｜MATH8202｜Theory of Partial Differential Equations偏微分方程理论 澳洲国立大学

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The course will discuss the three main classes of equations, elliptic, parabolic and hyperbolic. It is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered will include fundamental solutions, maximum principles, regularity (smoothness) of solutions, variational problems, Holder and Sobolev spaces.

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

(a) On the bounded domain $\Omega$ with smooth boundary, let $u$ be a solution of the problem
$$\Delta u+a_i(\mathbf{x}) \frac{\partial u}{\partial x_i}=f(\mathbf{x}), \quad \frac{\partial u}{\partial n}=0 \text { on } \partial \Omega .$$
Assume that $f \geq 0$ in $\Omega$. Show that $u$ is a constant and $f=0$.
(b) Show that problem (4.19) can have a solution only if
$$\int_{\Omega} f(\mathbf{x}) v(\mathbf{x}) d \mathbf{x}=0$$
for every solution $v$ of the “adjoint” equation
$$\Delta v-\frac{\partial}{\partial x_i}\left(a_i(\mathbf{x}) v\right)=0, \quad \frac{\partial v}{\partial n}-a_i(\mathbf{x}) n_i v=0 \text { on } \partial \Omega .$$
(c) Using techniques to be developed in later chapters, one can show that the condition)is also sufficient and that the solution space of is one-dimensional. Taking these facts for granted, show that solutions of are either non-negative or non-positive. Equations of the form are called Fokker-Planck equations and arise in statistical physics. Only non-negative solutions are physically meaningful.

Let $\Omega$ be a regular hexagon with side $a$. Let $\lambda \in \mathbb{R}$ be such that the equation $\Delta u+\lambda u=0$ with boundary condition $u=0$ has a nontrivial solution in $\Omega$. Give a lower bound for $\lambda$.

Verify that the function given by (4.22) is harmonic.

Prove the second claim in the subsection on subharmonic functions.
. Show that if $u$ is of class $C^2$ and subharmonic in the sense defined in this section, then $\Delta u \geq 0$.

Let the sequence $f_m$ be equicontinuous at each point of a compact set $S$. Show that it is uniformly equicontinuous on $S$.

Let $\Omega=\left{(x, y) \in \mathbb{R}^2 \mid 0<x^2+y^2<1\right}$. Prove that there is no solution to the Dirichlet problem $\Delta u=0$ in $\Omega, u(\mathbf{x})=1$ for $x^2+y^2=1$, $u(\mathbf{0})=0$. Hint: Show first that if there is a solution, then there is also a radially symmetric solution.

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