Hand in y our solutions un til F rida y , Ma y 18, 8. 30 am (PO b o x 183 in V3-128)
total points: 24
Prof. Dr. Moritz Kaßmann Fakultät für Mathematik
Sommersemester 2018 Universität Bielefeld
Partial Differential Equations III Exercise sheet V, May 12
Exercise V.1 (6 points)
Prove the following result from the lecture:
Proposition 2.9. LetX be a Banach space,(Tt) a strongly continuous semigroup and A the infinitesimal generator. Then the following assertions are equivalent
(i) (Tt) is uniformly continuous.
(ii) D(A) =X.
(iii) A is bounded.
In this case Tt=eAt.
Exercise V.2 (6 points)
(a) LetX= BUC(R),D={f ∈X|f0 ∈BUC(R)}and A:D→X, Af =f0. Show that A satisfies the three conditions of the Hille-Yosida theorem.
(b) LetX =W1,p(R) for 1≤p <∞and (Tt) be given byTtf(x) =f(x+t). Show that the infinitesimal generator is given by A:D→X, Af =f0, withD=W1,p(R). Discuss the applicability of the Hille-Yosida theorem.
Exercise V.3 (6 points)
Let Ω ⊂ Rd be a bounded open set with a smooth boundary. Assume aij ∈ C2(Ω) and(aij) uniformly positive definite. Let X =L2(Ω)and D=H01(Ω)∩H2(Ω). Define A:D→X byAu= div(a∇u).
(i) Show that Asatisfies the three conditions of the Hille-Yosida theorem.
(ii) Solvability of which parabolic problem follows?
(iii) Which generalizations of A can you consider easily?
Hint: You need to apply some regularity results from previous lectures.
Exercise V.4 (6 points)
LetA:D(A)⊂X→X be the infinitesimal generator of a contraction semigroup. Define D(A2) ={x∈D(A)|Ax∈D(A)} and A2x=A(Ax).
(a) Show
kAxk2 ≤4kxk kA2xk (x∈D(A2)). (b) Apply this result in the context of Exercise V.2 (b) and prove
kf0k2Lp≤4kfkLpkf00kLp (f ∈W2,p).
