課程名稱︰偏微分方程式二
課程性質︰數學研究所基礎課
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︰2015年04月28日(二)，10:20-12:10
考試時限：110分鐘
試題 :
Test 2 4/28/2015
1. 20% 1,p
Prove directly that if u∈Ｗ (0,1) for some 1 < p < ∞, then
1-(1/p) 1 p (1/p)
|u(x)-u(y)|≦|x-y| (∫|u'| dt) for a.e. x,y∈[0,1].
0
2. 20%
Assume 1 < p < ∞, and U is bounded.
1,p 1,p
(i) Prove that if u∈Ｗ (U), then |u|∈Ｗ (U).
1,p + - 1,p
(ii)Prove that u∈Ｗ (U) implies u, u ∈Ｗ (U), and
+ / Du a.e. on {u > 0}
Du =
\ 0 a.e. on {u≦ 0},
- / 0 a.e. on {u≧ 0}
Du =
\ -Du a.e. on {u < 0}.
+
(Hint: u = lim F_ε(u), for
ε→0
/ (z^2+ε^2)^(1/2) - ε if z≧ 0
F_ε(z):=
\ 0 if z < 0.)
1,p
(iii) Prove that if u∈Ｗ (U), then
Du = 0 a.e. on the set {u = 0}.
Integrate by parts to prove the interpolation inequality:
2 2 (1/2) 2 2 (1/2)
∫|Du| dx ≦ C(∫u dx) (∫|D u| dx)
U U U
∞ 2 1
for all u∈C (U). By approximation, prove this inequality if u∈H(U)∩H(U).
c 0
3. 20% 0
Fix α > 0 and let U = B(0,1). Show there exists a constant C, depending
only on n and α, such that
2 2
∫u dx ≦ C ∫|Du| dx,
U U
provided 1
|{x∈U|u(x)=0}|≧α, u∈H(U).
4. 20% 1 n
Let u∈C (R ). Prove that
c
1 |Du(y)|
-------- ∫|u(y)-u(x)|dy ≦ C ∫ ------------dy.
|B(x,r)| B(x,r) B(x,r) |y-x|^(n-1)
n
for any ball B(x,r)⊆R.
5. 20% 1,p n _n
Let u∈Ｗ (R ), u has compact support in R ,
+ n n-1 +
and Tu = 0 on ∂R = R , where T is the trace operator.
+
Prove that
p p-1 xn p-1
∫|u(x',x_n)| dx' ≦ C (x_n) ∫ ∫ |Du| dx'dt
R^(n-1) 0 R^(n-1)
for a.e. x_n > 0
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