臺灣大學數學系

 八十六學年度第二學期碩博士班資格考試試題

 微分方程式

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25 points each.

1. Solve the following PDEs.
   1. $e^xu_y - uu_x = 0, u(x,0) = -e^x$
   2. 　
      $$\begin{array}{l} u_{tt}-u_{xx}=x^2 \hbox{ in }\mathbb{R}\times [0,\infty) \\ u(x,0)=u_t(x,0)=0. \end{array}\hspace{8cm}$$
2. 　
   1. Show that for $n=3$ the general solution of $u_{tt}-c^2\triangle u=0$ with spherical symmetry about the orgin has the form
      
      $$u ={{F(r+ct)+G(r-ct)}\over{r}},\quad r=\vert x\vert$$
      
      with suitable $F,G$.
   2. Show that the solution with initial data of the form
      
      $$u=0, u_t=g(r)\ (g=\hbox{even function of} r)$$
      
      is given by
      
      $$u={1 \over{2cr}}\int_{r-ct}^{r+ct}\rho g(\rho)\,d\rho.$$
3. Let Ω be a bounded smooth domain in $\mathbb{R}^n$. Assume $u\in C^2(\bar\Omega)$ is a solution of
   
   $$\triangle u = 0\hbox{ in }\Omega, \, \, u= f \hbox{ on }\partial\Omega.$$
   
   
   1. Show that
      
      $$\int_{\Omega}\vert\nabla u\vert^2\,dx \le \int_{\Omega}\vert\nabla v\vert^2\,dx$$
      
      for any $v\in C^1(\bar\Omega)$ with boundary values $f$.
   2. Let $B_{\xi,\rho}=\{x:\vert x-\xi\vert<\rho\}\subset\Omega$. Show that
      
      $$u(\xi)={1\over{\omega_n\rho^{n-1}}} \int_{\partial B_{\xi,\rho}}u(x)\, ds_x,$$
      where $\omega_n\rho^{n-1}$ is the surface area of $\partial B_{\xi,\rho}$ and $ds_x$ is the surface element on $\partial B_{\xi,\rho}$.
4. Let $K(x,y,t)=(4\pi t)^{-{n \over 2}} e^{-{{\vert x-y\vert^2}\over{4t}}}$, $f$ be a continuous bounded function on $\mathbb{R}^n$ and
   $u(x,t)=\int_{\mathbb{R}^n}K(x,y,t)f(y)\,dy$.
   1. Show that
      
      $$K(x,0,s+t)=\int_{\mathbb{R}^n}K(x,y,t)K(y,0,s)\,dy$$
      for $s>0,t>0$.
   2. Show that $u$ satisfies $u_t=\triangle u$ for $t>0$ and

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