Hand in y our solutions un til F rida y , Ma y 3, 8.30 am (PO b o x 183 in V3-128)
total points: 20
Prof. Dr. Moritz Kaßmann Fakultät für Mathematik
Sommersemester 2018 Universität Bielefeld
Partial Differential Equations III Exercise sheet III, April 27
Exercise III.1 (5 points)
Prove the following theorem from the lecture:
Theorem 1.19. Let A:D⊂H → H be densely defined and symmetric. Then exactly one of the following assertions is true:
(i) σ(A)⊂R (ii) σ(A) =C
(iii) σ(A) ={λ∈C |Im(λ)≥0}
(iv) σ(A) ={λ∈C |Im(λ)≤0}.
If ρ(A)∩R6=∅, then σ(A)⊂R andA is self-adjoint.
Hint: Assume λ∈ σ(A), µ ∈ ρ(A) belong to the same open half-plane. Conclude that there is γ ∈σ(A) with Im(γ) 6= 0and a sequence (γn) ⊂ρ(A) with γn → γ such that kRγnk → ∞ for n→ ∞. Proceed from here.
Exercise III.2 (4 points)
Assume that A:D⊂H →H is a densely defined operator in a Hilbert spaceH. Assume that Ais closed and one-to-one and R(A) is dense in H. ThenA−1 :R(A)→H exists and is densely defined, and A∗ and (A−1)∗ both exist. Prove the following assertions:
(i) N(A∗) ={0}
(ii) R(A∗) is dense in H (iii) (A−1)∗ = (A∗)−1 (iv) A−1 is closed.
Exercise III.3 (5 points)
Assume A:H →H is a linear bounded self-adjoint operator which is one-to-one. Show thatR(A)is dense in H and thatA−1:R(A)→H is self-adjoint.
Exercise III.4 (6 points)
Let us look at an example of an integro-differential operatorLfrom Exercises I.4 and II.3.
For s∈(0,1) andd∈N, letc(s, d) be a positive constant satisfying lim
s%1 c(s,d) s(1−s) = ω4d
d−1, whereωd−1 denotes the(d−1)-dimensional volume of the sphereSd−1. Letνs be a Borel measure on Rd given by νs(dh) =c(s, d)|h|−d−2sdh. Let Ls be the operator defined in Exercise II.3 with ν replaced by νs. For u∈Cc∞(Rd) andx ∈Rd, the following holds true:
Lsu(x) = c(s, d) 2
Z
Rd
u(x+h)−2u(x) +u(x−h)
|h|−d−2sdh
Prove the following limit behavior:
s%1limLsu(x) = ∆u(x).
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