Sobolev Spaces
This chapter provides a comprehensive survey of the mathematical background of Sobolev spaces that is needed in the rest of the book. In addition to the standard notions, results, and calculus rules, various other useful topics, such as Green’s identity,
- PDF / 193,176 Bytes
- 13 Pages / 439.36 x 666.15 pts Page_size
- 46 Downloads / 205 Views
Sobolev Spaces
Abstract This chapter provides a comprehensive survey of the mathematical background of Sobolev spaces that is needed in the rest of the book. In addition to the standard notions, results, and calculus rules, various other useful topics, such as Green’s identity, the Poincaré–Wirtinger inequality, and nodal domains, are also discussed. A careful distinction between various properties of Sobolev functions is made with respect to whether they are defined on a one-dimensional interval or a multidimensional domain. Bibliographical information and related comments can be found in the Remarks section.
1.1 Sobolev Spaces In this chapter we gather some basic results from the theory of Sobolev spaces that we will need in the sequel. We simply state the results, and for their proofs we refer to one of the standard books on the subject mentioned in the final section of this chapter. Sobolev spaces are the main tool in the modern approach to the study of nonlinear boundary value problems. We start by fixing our notation. For a measurable set E ⊂ RN (N ≥ 1) and 1 ≤ p ≤ +∞, we denote by (L p (E, RM ), · p ) the Banach space of measurable functions u : E → RM (M ≥ 1) for which the quantity
u p :=
⎧ 1 p ⎪ ⎨ |u(x)| p dx if 1 ≤ p < +∞, E
⎪ ⎩ ess sup |u(x)| E
if p = +∞
is finite. Hereafter, | · | denotes the Euclidean norm of RM , which coincides with the absolute value if M = 1. We abbreviate L p (E) = L p (E, R). Let Ω ⊂ RN be an open set. Recall that f : Ω → R is locally integrable if, for every K ⊂ Ω compact, f ∈ L1 (K). The space of locally integrable D. Motreanu et al., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, DOI 10.1007/978-1-4614-9323-5__1, © Springer Science+Business Media, LLC 2014
1
2
1 Sobolev Spaces
1 (Ω ). Also, set C∞ (Ω ) = {ϑ ∈ C∞ (Ω ) : functions on Ω is denoted by Lloc c ϑ has compact support in Ω }. A multi-index is an N-tuple α = (αk )Nk=1 ∈ (N0 )N , where N0 = N ∪ {0}. For an index k ∈ {1, . . . , N}, let Dk = ∂∂x denote the kth partial derivation operator for k differentiable real functions on Ω . A multi-index gives rise to a classical differential operator of higher order, Dα = Dα1 1 · · · DαNN , defined on smooth functions. 1 (Ω ). We say that v Definition 1.1. Let α = (αk )Nk=1 be a multi-index and u, v ∈ Lloc α is the weak (or distributional) derivative of u, denoted by D u, if
Ω
u Dα ϑ dx = (−1)α1 +···+αN
Ω
v ϑ dx for all ϑ ∈ Cc∞ (Ω ).
If α = (0, . . . , 0), then we write Dα u = u. Remark 1.2. If u is smooth enough to have a classical continuous derivative Dα u, then we can integrate by parts and conclude that the classical derivative coincides with the weak one. Of course, the weak derivative may exist without having the existence of the classical derivative. To define the weak derivative Dα u, we do not need the existence of derivatives of smaller order. Moreover, the weak derivative 1 (Ω ), is defined up to a Lebesgue-null set, and it is Dα u, being an element of Lloc unique. Definition 1.3. Let Ω ⊂ RN be an
Data Loading...
