課程名稱︰偏微分方程式一
課程性質︰研究所基礎課
課程教師︰林太家
開課學院：理學院
開課系所︰數學系、數學研究所、應用數學科學研究所
考試日期︰2014年10月14日(二)，10:20-12:10
考試時限：110分鐘
是否需發放獎勵金：是
試題 :
Test 1 10/14/2014
1. (20%) Φ(x) = -log|x|/(2π) for x∈R^2, x≠0. Prove that
-△u = f in R^2,
where f ∈C(∞,0)(R^2) and u(x) = ∫ Φ(x-y)f(y)dy for x ∈R^2.
R^2
_ 1
2. (20%) Let u(r) = ---------- ∫ u(x)dS_x for r > 0
r^(n-1) ∂B(0,r)
and u ∈C^2(R^n). Prove that
_ ___
△u = △u for r > 0.
_
Note that u is radially symmetric in R^n
but u may NOT be radially symmetric in R^n.
3. (20%)
Prove that Laplace's equation △u = 0 is rotation invariant; that is, if
O is an orhogonal n ×n matrix and we define
v(x) := u(Ox) (x∈R^n),
then △v = 0.
4. (20%)
Let U^+ denote the open half-ball {x∈R^n | |x|<1, x_n > 0}.
_
Assume u∈C(U^+) is harmonic in U+, with u = 0 on ∂U^+ ∩{x_n = 0}.
Set
v(x) := u(x) if xn≧0
-u(x1,...,x_(n-1),-x_n) if xn＜0
for x∈U=B^o(0,1). Prove v is harmonic in U.
5. (20%)
_
We say v∈C^2(U) is subharmonic if
-△v≦0 in U.
(a) Prove for subharmonic v that
1
v(x) ≦ ---------- ∫vdy for all B(x,r)⊂U.
|B(x,r)| B(x,r)
(b) Prove that therefore max v = max v.
U ∂U
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