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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.