Repetition questions for Partial differential equations with
distribution theory.
The Course is defined by Renardy Rogers chapters 1,2,3 and 5, and lecture notes. To
receive pass on the oral exam you should do well in the basic questions below. To receive
a high grade you should do well on most of the questions below.
Basic questions
1. Give the definition of the space of test functions C0∞ (Ω) on an open set Ω in Rn .
2. What does it means for a sequence of test functions to converge?
3. Define 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. Define what it means for a sequence of distributions to converge.
6. Define the derivative of a distribution.
7. Compute the derivative (using the definition) of the Heaviside step function.
8. Define the order of a distribution.
9. Give a definition of convolution of a distribution and a testfunction in Rn .
10. Define 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 differential operator L(D) with
constant coefficients?
13. Compute a fundamental solution of the standard differentiation operator
the line R.
d
dx
on
14. Give the definition 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?
16. State the local in time existence and uniqueness of solutions to the ordinary differential 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 :=
principal symbol?
P
|α|≤m aα (x)D
α u?
What is the
19. What does it mean for a second order scalar PDE in Rn to be elliptic, parabolic,
hyperbolic or ultrahyperbolic?
20. Give the definition of a characteristic surface for a linear scalar PDE.
21. Give the definition of a characteristic surface for a nonlinear scalar PDE.
22. What does this definition mean for the characteristic curves for the scalar conservation law ∂t u + ∂x (f (u)) = 0?
23. Describe a procedure for solving first-order quasilinear PDEs.
24. What does the Rankine-Hugoniot condition say for the scalar conservation law
∂t u + ∂x (f (u)) = 0?
25. State the Cauchy-Kowalevski theorem
26. Give the definition of Riemann invariants for a strictly hyperbolic system of conservation laws.
27. Define the concepts Riemann problem, shock wave and rarefaction wave.
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. Define 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?
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
differential 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.
14. Show that a smooth solution to the scalar conservation law ∂t u + ∂x (f (u)) = 0is
constant along a characteristic curve.
15. Show that a characteristic curve for the scalar conservation law ∂t u+∂x (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 discontinuityfor the scalar conservation law ∂t u + ∂x (f (u)) = 0.