Sobolev spaces on warped products

Valuations on Sobolev Spaces

All affinely covariant convex-body-valued valuations on the Sobolev space W (R) are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function f ∈W (R) the unit ball of its optimal Sobolev norm. 2000 AMS subject classification: 46B20 (46E35, 52A21,52B45) Let ‖ ·‖ denote a norm on...

Sobolev spaces on graphs

max(u,v)∈E |f(u)− f(v)| if p =∞. If G is connected, then the only functions f satisfying ||f ||E,p = 0 are constant functions, so || · ||E,p is a norm on each linear space of functions on VG which does not contain constants. Usually we shall consider the subspace in the space of all functions on VG given by ∑ v∈V f(v)dv = 0. The obtained normed space will be called a Sobolev space on G and will...

Warped Products and Conformal Boundaries of Cat(0)-spaces

We discuss the conformal boundary of a warped product of two length spaces and provide a method to calculate this in terms of the individual conformal boundaries. This technique is then applied to produce CAT(0)-spaces with complicated conformal boundaries. Finally we prove that the conformal boundary of an Hadamard n-manifold is always simply connected for n ≥ 3, thus providing a bound for the...

Lecture Notes on Sobolev Spaces

We denote by Lloc(IR) the space of locally integrable functions f : IR 7→ IR. These are the Lebesgue measurable functions which are integrable over every bounded interval. The support of a function φ, denoted by Supp(φ), is the closure of the set {x ; φ(x) 6= 0} where φ does not vanish. By C∞ c (IR) we denote the space of continuous functions with compact support, having continuous derivatives ...

Sobolev Spaces