課程名稱︰偏微分方程式二
課程性質︰數學研究所必選修
課程教師︰陳逸昆
開課學院：理學院
開課系所︰數學研究所
考試日期︰2018年06月26日(二)
考試時限：10:20-12:10，共計110分鐘
試題 :
　　　　　　　　　　　　　 PDE, Spring 2018
　　　　　　　　　　　　　　 　Final Exam
DEP. _______________　　NAME_________________　ID NUMBER___________________
1. Let 2 i+j
L(u) = - Σ [i+j+(-1) ]u .
i,j=1 xi xj
(a.) Show that L is uniformly elliptic. (15%)
2
(b.) Assume Ω be a bounded open set in |R. Show that the boundary value
problem
L(u) = f in Ω,
u = 0 on ∂Ω,
2 1
where f∈L (Ω), has a weak solution in H (Ω). (10%)
0
1
2. Assume U is a bounded connected open set with C boundary.
1
A function u∈H (U) is weak solution of Neumann's problem
-Δu = f in U
∂u