Study questions for the course Partial dierential equations
with distribution theory.
The Course is dened by Renardy Rogers (RR) Chapters 1,2,3 and 5, and
lecture notes handed out during the course. To receive a pass on the oral
exam you should do well on the basic questions below. To receive a high
grade you should do well on all of the questions below.
Basic questions
1. Give the denition of the space of test functions D(Ω) on an open set
Ω in Rn .
2. What does it means for a sequence of test functions to converge?
3. Dene the space of distributions D0 (Ω) on an open set Ω in Rn .
4. In what way can we identify a continuous function (or a local L1 function) with a distribution?
5. Dene what it means for a sequence of distributions to converge.
6. Dene the derivative of a distribution.
7. Compute the derivative (using the denition) of the Heaviside step
function.
8. Dene the order of a distribution.
9. Give a denition of convolution of a distribution and a testfunction in
Rn .
10. Dene the support of a distribution.
11. What does a distribution on Rn with support in {0} look like?
12. What is a fundamental solution on Rn of a linear dierential operator
L(D) with constant coecients?
13. Compute a fundamental solution of the standard dierentiation operd
ator dx
on the line R.
14. Give the denition of a Green's function for the Dirichlet boundary
value problem for the Laplace-operator on an open bounded domain
in Rn .
15. What is the relation between a fundamental solution and a Green's
function?
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16. State the local in time existence and uniqueness of solutions to the
ordinary dierential equation ẋ(t) = f (x(t)), x(0) = x0 when f is
Lipschitz continuous.
17. State the inverse function theorem.
18. What is the symbol of an operator L(x, D)u :=
What is the principal symbol?
P
|α|≤m aα (x)D
α u?
19. What does it mean for a second order scalar PDE in Rn to be elliptic,
parabolic, hyperbolic or ultrahyperbolic?
20. Give the denition of a characteristic surface for a linear scalar PDE.
21. Give the denition of a characteristic surface for a nonlinear scalar
PDE.
22. What does this denition mean for the characteristic curves for the
d
scalar conservation law ut + dx
(f (u)) = 0?
23. Describe a procedure for solving rst-order quasilinear PDEs.
24. What does the Rankine-Hugoniot condition say for the scalar conserd
(f (u)) = 0?
vation law ut + dx
25. State the Cauchy-Kowalevski theorem
26. Give the denition of Riemann invariants for a strictly hyperbolic system of conservation laws.
27. Dene the concepts
wave.
Riemann problem, shock wave and rarefaction
Advanced questions
1. Prove that there exist testfunctions on Rn .
2. Prove that there exist distributions on R of order k ∈ N.
3. Prove that the Dirac delta distribution can not be represented by a
continuous function.
4. Dene homogeneity of a distribution in Rn .
5. Prove that the Dirac delta distribution is homogeneous on Rn . What
is the degree?
6. Prove that a derivative of a homogeneous distribution is again homogeneous. What happens with the degree?
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7. Prove that the convolution of a testfunction and a distribution in Rn
is a smooth function.
8. Prove that if u ∈ D0 (R) and u0 = 0 then u is constant.
9. Prove that a distribution on Rn with support in {0} is a sum of derivatives of the Dirac distribution.
10. Compute a fundamental solution for the Laplace-operator on Rn .
11. Prove the local in time existence and uniqueness of solutions to the
ordinary dierential equation ẋ(t) = f (x(t)), x(0) = x0 when f is
Lipschitz continuous, for instance using the BFPT.
12. Sketch a proof for the Cauchy-Kowalevski theorem.
13. State Holmgren's uniqueness theorem and describe the idea of the
proof.
d
14. Show that a smooth solution to the scalar conservation law ut + dx
(f (u)) =
0 is constant along a characteristic curve.
15. Show that a characteristic curve for the scalar conservation law ut +
d
dx (f (u)) = 0 is a straight line.
16. Prove that the Rankine-Hugoniot condition holds for a weak solution
that is a classical solution on "both sides of the jump discontinuity"
d
for the scalar conservation law ut + dx
(f (u)) = 0.
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