Hand in y our solutions un til F rida y , June 1, 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 VII, May 25
Exercise VII.1 (10 points)
At the end of Section 2 of the lecture, a diagram was presented that connects the three objects
(i) (Tt)t>0 - a strongly continuous contraction semigroup (ii) (Gλ)λ>0 - a strongly continuous contraction resolvent (iii) A:D⊂X →X - a densely defined (closed) operator with
(a) (0,∞)⊂ρ(A),
(b) k(λI−A)−1k ≤λ−1 ∀λ >0.
Formulate six assertions about the respective connection (existence/uniqueness). Provide the proofs or refer to theorems resp. proofs given in the lecture.
Exercise VII.2 (4 points)
LetE :D×D→Rbe a symmetric form.
(i) Show that E is closable if and only if it has a closed extension.
(ii) Show that the condition
(un)⊂D,(un, un)→0 asn→ ∞
=⇒ ∀v∈D:E(un, v)→0 implies that E is closable.
Exercise VII.3 (6 points)
LetΩ⊂Rd be a domain andkbe a positive Radon measure on Ω. Let J be a positive Radon measure onΩ×Ω\diagsuch that for every compact set K and every open set D withK ⊂D⊂Ω
Z
K×K\diag
|x−y|2J(dxdy)<∞, J(K×(Ω\D))<∞.
(a) SetD(E) =Cc∞(Ω)and E(u, v) =
Z
Ω×Ω\diag
u(y)−u(x)
v(y)−v(x)
J(dxdy) + Z
Ω
u(x)v(x)k(dx).
Show that the formE is a closable Markov form onL2(X, m), wheremis any positive Radon measure on Ωwithsupp(m) = Ω.
(b) Provide a nontrivial example of J.
(c) Choose L be a nonlocal operator from Exercise II.3 or Exercise III.4. For whichJ and k, and for which functionsu andv does
− Z
Rd
(Lu)(x)v(x)dx=E(u, v)
hold true?
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