Function Spaces
In this chapter we introduce function spaces that will be relevant to the subsequent developments in this monograph. The function spaces to be discussed include spaces of continuous and continuously differentiable functions, smooth functions, Lebesgue and
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Function Spaces
In this chapter we introduce function spaces that will be relevant to the subsequent developments in this monograph. The function spaces to be discussed include spaces of continuous and continuously differentiable functions, smooth functions, Lebesgue and Sobolev spaces, associated with an open bounded domain in Rd . In order to treat time-dependent problems, we also introduce spaces of vector-valued functions, i.e., spaces of mappings defined on a time interval Œ0; T with values in a Banach or a Hilbert space.
2.1 Spaces of Smooth Functions Spaces of continuously differentiable functions. Everywhere in this section ˝ denotes an open bounded subset of Rd and x D .x1 ; : : : ; xd / will represent a generic point of ˝. We define C.˝/ to be the set of all continuous functions from ˝ to R. The set C.˝/ forms a linear space under the usual addition and scalar multiplication. Similarly, the notation C.˝/ is used for the space of real-valued functions continuous on ˝. Since ˝ is a bounded set, the space C.˝/ consists of functions which are uniformly continuous on ˝ and it is a Banach space with the norm kvkC.˝/ D sup jv.x/j D max jv.x/j: x2˝
x2˝
It is clear that C.˝/ C.˝/ with the proper inclusion. Indeed a simple onedimensional example of function which belongs to C.˝/ and does not belong to C.˝/ is given by taking v.x/ D 1=x on ˝ D .0; 1/ R. We introduce some space of continuously differentiable functions which can be endowed with a Banach space structure. To this end we adopt the following notion of multi-indices which is useful as a compact notation for partial derivatives.
S. Mig´orski et al., Nonlinear Inclusions and Hemivariational Inequalities, Advances in Mechanics and Mathematics 26, DOI 10.1007/978-1-4614-4232-5 2, © Springer Science+Business Media New York 2013
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2 Function Spaces
A multi-index m is an ordered d -tuple of integers m D .m1 ; : : : ; md /, mi 0 for i 2 f1; : :P : ; d g. The length jmj of the multi-index is the sum of the components of m, jmj D diD1 mi . Let vW ˝ ! R, Di D @=@xi and @jmj v @m1 CCmd v D m @x m @x1 1 @x2m2 @xdmd
D m v D D1m1 Ddmd v D
with D 0 v D v. Let k 2 N. We denote by C k .˝/ the vector space of all functions v which, together with all their partial derivatives D m v of orders jmj k, are continuous on ˝. Here and everywhere in this book, for k D 0, we set C 0 .˝/ D C.˝/ and C 0 .˝/ D C.˝/. The set of k-times continuously differentiable functions on ˝ is recursively defined by C k .˝/ D fv 2 C k1 .˝/ j D m v 2 C.˝/ for all m such that jmj D kg: The set C k .˝/ is a Banach space with the norm kvkC k .˝/ D
X
kD m vkC.˝/ :
jmjk
We also have a proper inclusion C k .˝/ C k .˝/. The spaces of infinitely differentiable functions are defined by C 1 .˝/ D
\
C k .˝/;
k2N0
C 1 .˝/ D
\
C k .˝/:
k2N0
Given ˝1 , ˝2 Rd , we recall that ˝1 ˝2 means that ˝ 1 ˝2 and ˝ 1 is compact in Rd . For a function vW ˝ ! R, its support is defined by supp v D fx 2 ˝ j v.x/ 6D 0g: We say that v has a compact support if supp v
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