Sobolev Spaces

Frontmatter

Introduction

Abstract

In [229–231] Sobolev proved general integral inequalities for differentiable functions of several variables and applied them to a number of problems of mathematical physics. Sobolev introduced a notion of the generalized derivative and considered the Banach space W _p ^l (Ω) of functions in L _p (Ω), p ≥ 1, with generalized derivatives of order l summable of order p. In particular, using his theorems on the potential type integrals as well as an integral representation of functions and the properties of mollifications, Sobolev established the imbedding of W _p ^l (Ω) into L _q (Ω) or C(Ω) under certain conditions on the exponents p, l, q.

Vladimir G. Maz’ja

Chapter 1. Basic Properties of Sobolev Spaces

Abstract

Let Ω be an open subset of n-dimensional Euclidean space R ⁿ = {x}.

Vladimir G. Maz’ja

Chapter 2. Inequalities for Gradients of Functions that Vanish on the Boundary

Abstract

The present chapter deals with necessary and sufficient conditions for the validity of certain estimates for the norm \({\left\| u \right\|_{{L_q}(\Omega ,\mu )}}\) , where u ∈ D(Ω) and μ is a measure in Ω.

Vladimir G. Maz’ja

Chapter 3. On Summability of Functions in the Space L 1 1 (Ω)

Abstract

The present chapter contains conditions on Ω which are necessary and sufficient for the imbedding operator L ₁ ¹ (Ω) → L _q (Ω) to be continuous or compact.

Vladimir G. Maz’ja

Chapter 4. On Summability of Functions in the Space L p 1 (Ω)

Abstract

Let Ω be an open set in R ⁿ . Let F and G denote bounded closed (in Ω) and open subsets of Ω, respectively, F ⊂ G.

Vladimir G. Maz’ja

Chapter 5. On Continuity and Boundedness of Functions in Sobolev Spaces

Abstract

If a domain Ω has the cone property, then by the Sobolev theorem any function u in W _p ^l (Ω), pl > n, coincides almost everywhere with a continuous function in Ω, and

$${\left\| u \right\|_{{L_{\infty (\Omega )}}}}C{\left\| u \right\|_{w_p^l(\Omega )}},$$

where the constant C does not depend of u.

Vladimir G. Maz’ja

Chapter 6. On Functions in the Space B V(Ω)

Abstract

In the present chapter, written together with Ju.D. Burago, we consider the space B V(Ω) of functions whose generalized derivatives are measures in the open set Ω ⊂ R ⁿ .

Vladimir G. Maz’ja

Chapter 7. Certain Function Spaces, Capacities and Potentials

Abstract

The present chapter is of an auxiliary nature. Here we collect (mostly without proofs) the results of function theory which are applied later or are related to the facts used in the sequel. First we discuss the theorems on spaces of functions having derivatives of arbitrary positive order (§ 7.1). The theory of these spaces is essentially presented in monographs (cf. Stein [237], Peetre [210], Nikol’skiĭ [202], Besov, Il’in and Nikol’skiĭ [27], Triebel [244, 245]) though in some cases the reader interested in the proofs will have to refer to the original papers. Section 7.2 deals with the properties of capacities and nonlinear potentials. Unfortunately, this material is dispersed throughout the journals. However, an attempt at its systematic exposition would lead to an inadmissible enlargement of the volume. In any case, the presented statements are sufficient for our further considerations.

Vladimir G. Maz’ja

Chapter 8. On Summability with Respect to an Arbitrary Measure of Functions with Fractional Derivatives

Abstract

According to Corollary 2.3.3, for q ≥ p the inequality

$$\parallel u{\parallel _{{L_q}(\mu ,{R^n})}} \leqslant A\parallel \nabla u{\parallel _{{L_p}({R^n})}},\;\;u \in C_0^\infty ,$$

(1)

, follows from the “isoperimetric” inequality

$${(\mu (E))^{p/q}} \leqslant {p^{ - p}}{(p - 1)^{p - 1}}{A^p}\;cap(E,w_p^1)$$

.

Vladimir G. Maz’ja

Chapter 9. A Variant of Capacity

Abstract

In 7.2.1 we introduced the capacity cap(e, S _p ^l of a compactum e ⊂ Rⁿ for any one of the spaces S _p ^l = H _p ^l , h _p ^l , B _p ^l etc.

Vladimir G. Maz’ja

Chapter 10. An Integral Inequality for Functions on a Cube

Abstract

Let Q _d be an open n-dimensional cube with edge length d and with sides parallel to coordinate axes. Let p ≥ 1, and k, l be integers, 0 ≤ k ≤ l. We denote a function in W _l ^p (Q _d ), p ≥ 1, by u.

Vladimir G. Maz’ja

Chapter 11. Imbedding of the Space into Other Function Spaces

Abstract

If n > pl, p>1 or n ≥ l, p = 1 then for all u ∈ D(Ω) the Sobolev inequality

$$\parallel u{\parallel _{{L_p}(\Omega )}} \leqslant C\parallel {\nabla _l}u{\parallel _{{L_p}(\Omega )}},$$

(1)

, where q=pn/(n-pl), is valid.

Vladimir G. Maz’ja

Chapter 12. The Imbedding

Abstract

In this chapter we denote by L̊ _p ^l (Ω, v) the completion of D(Ω) with respect to the metric

$$\parallel {\nabla _l}u{\parallel _{{L_p}(\Omega )}} + \parallel u{\parallel _{{L_p}(\Omega ,v)}},$$

, where p > 1, Ω is an open set in R ⁿ and v is a measure in Ω.

Vladimir G. Maz’ja

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