課程名稱︰偏微分方程式二
課程性質︰數學系選修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/05/04
考試時限（分鐘）：120
試題 :
Each of the following problems counts 25 points.
1. State the Fredholm alternative for identity-compact operators on Hilbert
space.
2. State the Lax-Milgram theorem and prove it.
3. Suppose the Hilbert space H is of infinite dimension and K: H → H is compact
and linear. Show that
(1) $0 \in \sigma_s(K)$,
(2) $\sigma_s(K)\backslash\{0\} = \sigma_p(K)\backslash\{0\}$, and
(3) $\sigma_s(K)\backslash\{0\}$ is finite, or a sequence tending to 0.
4. Assume U is connected, smooth and bounded. A function $u \in H^1(U)$ is a
weak solution of Neumann's problem
\begin{equation}\label{eq1}
\begin{cases}
-\Delta u = f & \text{in }U\\
\frac{\partial u}{\partial\nu} = 0 & \text{on }\partial U
\end{cases}
\end{equation}
if
\[\int_U Du \cdot Dv dx = \int_U fv dx\]
for all $v \in H^1(U)$. Suppose $f \in L^2(U)$. Prove (\ref{eq1}) has a weak
solution if and only if
\[\int_U f dx = 0.\]
(Hint: You might want to use Lax-Milgram theorem and Fredholm alternative to
conclude the result. You have to show the reasonings step by step.)