MATH 655 Supplementary Homework
Fall 2012
In all of these problems P (D) denotes a constant coefficient linear partial differential
operator of order m ≥ 1 in RN
8. If N = 1 show that P (D) is elliptic.
9. If N > 1, m = 1 and the coefficients are real, show that P (D) is not elliptic.
10. If P (D) is elliptic, show that lim|y|→∞ |P (y)| = ∞ and that
N (P ) = {y ∈ RN : P (y) = 0}
is compact.
11. If f ∈ C0∞ (RN ) show that the equation P (D)u = f has at most one solution
u ∈ C0∞ (RN ). Show by example that this is false if we only require u ∈ C ∞ (RN ) and
clarify why your proof above doesn’t work in this case.