Hand in y our solutions un til F rida y , June 15, 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 IX, June 11
Exercise IX.1 (20 points)
Assume0< s <1 andp, q, c≥1. The following assertion is sometimes called Poincaré- Sobolev inequality of fractional order:
kukLq(Rd)≤c ¨
RdRd
u(y)−u(x)
p
|y−x|d+sp dy dx1/p
u∈Cc∞(Rd)
(1)
(i) Show that (1) may hold only for a very special relation betweenp andq. Compute q as a function of p. Explain how your answers would change ifRdwere replaced by an open bounded subset of Rd.
(ii) Prove (1) using the following ansatz:
|u(x)| ≤
Br(x)
|u(x)−u(y)|dy+
Br(x)
|u(y)|dy ,
wherex is an arbitrary point inRd andr >0 needs to be chosen in a clever way.
(iii) Deduce an embedding ofHs(Ω)intoLq(Ω)from (1) for open subsetsΩ⊂Rd. (iv) Find this embedding result in the literature, provide the reference and list the
ingredients of its proof.
