Hand in y our solutions un til F rida y , April 20, 8.30 am (PO b o x 183 in V3-128)

Hand in y our solutions un til F rida y , April 20, 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 I, April 13

Exercise I.1

(a) Let A:D⊂X →X be closed and µ∈ρ(A). Show that the following two assertions are equivalent:

(i) The embedding(D,k · k_D),→X is compact.

(ii) The resolvent R_µ:X→X is compact

(b) AssumeA:D⊂X→X,λn∈ρ(A) for n∈Nand λ= lim

n→∞λn. Assume sup

n

kR_λnk<∞.

Show thatλ∈ρ(A).

Exercise I.2

Let H be a complex separable Hilbert space. Let A be a dissipative operator on H.

Assume that there isλ∈C+ ={ω∈C|<(ω)>0} such that(λI−A)is surjective. Show that for everyµ∈C+

µ∈ρ(A), kR_µk ≤ <(µ)⁻¹.

Exercise I.3

Let X =C∞(R^d) = {f ∈C(R^d)| lim

|x|→∞f(x) = 0}, equipped with the supremum norm.

LetD=C_c²(R^d). Show that∆ :D→X is dissipative.

Exercise I.4

Letν be a Borel measure onR^d withν({0}) = 0,ν(A) =ν(−A) for every setA, and Z

R^d

min(1,|h|²)ν(dh)<∞.

(a) Provide examples for ν such that the resp. condition is satisfied:

(i) ν(R^d) = +∞, (ii) ν(R^d)<+∞, (iii) R

R^dmin(1,|h|)ν(dh) = +∞,

(iv) ν is not absolutely continuous with respect to the Lebesgue measure andsupp(ν) is unbounded.

(b) Provide conditions, as weak as possible, on a measurable functionf :R^d→R such that, givenx∈R^d, the expression

ε→0lim Z

R^d\Bε(0)

f(x+h)−f(x) ν(dh)

is well defined and, as a function ofx, is continuous.

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