PhD Qualify Exam, PDE, Sep. 21, 2001
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1.[10%] Verify that u(x, t) = 1 2c
∫ t 0
∫ x+ct−cs
x−ct+cs f (y, s) dyds is the solution of
utt + c2uxx = f
u(x, 0) = ut(x, 0) = 0. (1)
2.[10%] Let ϕ(x) be a bounded continuous function for −∞ < x < ∞. Define the function u(x, t) = 1
√4π
∫ ∞
−∞e−p2/4ϕ(x − p√
t) dp. Assume that the function u is C∞ for −∞ < x < ∞ and 0 < t < ∞. (Take it for granted.) Show that
tlim→0+u(x, t) = ϕ(x).
3.(a)[10%]Let D be an open disk in R2 of radius r and center Q. Suppose that u is harmonic in D and that
∫ ∫
D|u|2dxdy = K < ∞. Prove that
|u(Q)| ≤ 1 r
(K
π
)1
2
.
Note: You may not assume that u is continuously extendable to the closure of D.
Hint: Consider
∫ ∫
[u− u(Q)]2 dxdy.
(b)[10%] Suppose that u is harmonic on all of R2, and that
∫ ∫
R2|u|2dxdy <
∞. Prove that u is constant.
4.[10%] State a method of finding the Green’s function for the eighth of a ball,
D = {x2 + y2 + z2 < a2 : x > 0, y > 0, z > 0}.
5. Let u be a solution of the wave equation in all of R3× R. Suppose that a > 0 and that u(x, 0) = ut(x, 0) = 0 for |x| ≥ a.
(a)[10%] Show that u(x, t) = 0 in the double cone |x| ≤ |t| − a for |t| ≥ a.
(b)[10%] Show that there is a constant C > 0 such that
∫
R3u2(x, t) dx ≤ C, for all t > 0.
(Hint: Show that there is a finite energy solution of wtt−∆w = 0 such that wt = u.) 6. Let Ω be an open bounded subset of Rn, with smooth boundary. Consider the boundary value problem
−∆u + λu = f in Ω
∂u
∂n + γu = 0 on ∂Ω (2)
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where γ > 0.
(a)[5%] Give a weak formulation of this problem.
(b)[5%] Suppose that there is a weak solution for the problem (2), which is smooth in Ω. Show that this solution satisfies (2) in the usual (strong) sense.
( Recall: We say that v is a weak derivative of u if (v, ϕ) = −(u, ϕ′) for all test function ϕ.)
7. Suppose that g ∈ C1(R) and that M ≡ sup
R |g(x)| < ∞. Consider the initial
value problem
ut + u2ux = 0 for x ∈ R, t > 0
u(x, 0) = g(x) for x ∈ R. (3)
(a)[10%] Show that a C1 solution u(x, t) satisfies |u(x, t)| ≤ M for as long as the solution exists.
(b)[10%] Suppose that at some x0 ∈ R, g(x0)g′(x0) < 0. Show that a C1 solutioin breaks down in finite time. (Hint: study the behaviour of ux along char- acteristics.)
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