Sobolev Spaces on Domains

Frontmatter

Notation and basic inequalities

Victor I. Burenkov

Chapter 1. Preliminaries

Victor I. Burenkov

Chapter 2. Approximation by infinitely differentiable functions

Abstract

Let A _δ be a mollifier with the kernel ω defined in Section 1.1. We start by studying the properties of A _δ in the case of Sobolev spaces.

Victor I. Burenkov

Chapter 3. Sobolev’s integral representation

Victor I. Burenkov

Chapter 4. Embedding theorems

Abstract

The main aim of this chapter is to prove various inequalities related to those of the form

$$ {\left\| {D_w^\alpha f} \right\|_{{L_q}(\Omega )}} \leqslant {c_1}{\left\| f \right\|_{w_p^\iota (\Omega )}},$$

where α ∈ ℕ₀ ⁿ, |α| < l (D _w ⁰ f ≡ f) and c ₁ > 0 does not depend on f.

Victor I. Burenkov

Chapter 5. Trace theorems

Abstract

Let f ∈ L ₁ ^loc (ℝ ⁿ ) where n > 1. We would like to define the trace tr f ≡ tr_{R m} f ≡ f |_{ℝ m} of the function f on ℝ ^m where 1 ≤ m < n.

Victor I. Burenkov

Chapter 6. Extension theorems

Abstract

The main aim of this chapter is to prove that under sertain assumptions on an open set Ω ⊂ℝ ⁿ there exists an extension ¹ operator

$$ T:W_p^\iota (\Omega ) \to W_p^\iota ({{\Bbb R}^n}),$$

which is linear and bounded. The existence of such an operator ensures that a number of properties of the space W _p ^l (ℝ ⁿ ) are inherited by the space W _p ^l (Ω). Examples have been given in Section 4.2 (Remark 11 and the proof of Theorem 3) and Section 4.7 (Corollaries 20, 24 and the second proof of Theorem 13).

Victor I. Burenkov

Chapter 7. Comments

Abstract

The first exposition of the theory of Sobolev spaces was given by S.L. Sobolev himself in his book [134] and later in his other book [135].

Victor I. Burenkov

Backmatter