课程名称︰偏微分方程式一
课程性质︰数学系选修
课程教师︰夏俊雄
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/10/06
考试时限（分钟）：50
试题 :
0 ＿
1. Let's define the subharmonic function as follows: A function f(x) ∈ C (Ω)
is called a subharmonic function if for any interior point x ∈ Ω, there exists
a positive number ρ (We don't exclude the possibility that this positive number
might depend on x.) such that for any positive number r < ρ we have
1
f(x) ≦ ────∫ f(y) dy.
|B (x)| B (x)
r r
Now, you show
(A) (15%) If Ω is an open bounded simply connected set, the subharmonic
＿
functions defined on Ω satisfy the strong maximum principle.
＿
(B) (25%) Show that it is impossible for a subharmonic function f defined on Ω
that you can find an interior point x ∈ Ω and a radius r satisfying
1
f(x) > ────∫ f(y) dy.
|B (x)| B (x)
r r
Hence, this means that for a subharmonic function f and any B (x) ⊂ Ω, we
r
always have
1
f(x) ≦ ────∫ f(y) dy.
|B (x)| B (x)
r r
m
(C) (20%) Show that if {f } are subharmonic functions, then so is
i i=1
f(x) = max {f (x)}.
1≦i≦m i
2. (40%) Solve the following two differential equations:
╭ uu + u = 1,
│ x y
╯
│ 1
╰ u(x, x) = ─x.
2
╭ u - u = 0,
│ tt xx
│
╯ u (x, 0) = x,
│ t
│ x
╰ u(x, 0) = e .