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https://math.stackexchange.com/questions/2574640/class-of-compact-support-functions-is-dense-in-wk-p-mathbb-rn
I suppose it is a straightforward application of the definition of convergence in Sobolev spaces combined with a pointwise estimate for the derivatives, arising when applying the Leibniz rule. Let $\phi \in C^\infty (\mathbb R)$ such that
https://www.encyclopediaofmath.org/index.php/Sobolev_space
Its elements are usually generalized functions, that is, linear functionals $(f,\phi)$ on infinitely-differentiable functions $\phi$ with compact support in $\Omega$.
https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf
Sobolev spaces In this chapter we begin our study of Sobolev spaces. The Sobolev space is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces…
https://www.math.uh.edu/~rohop/spring_11/downloads/Chapter2.pdf
Chapter 2 Sobolev spaces In this chapter, we give a brief overview on basic results of the theory of Sobolev spaces and their associated trace and dual spaces. 2.1 Preliminaries ... compact support in ›). Finally, C1(›) stands for the set of functions with continuous partial
https://www.math.uci.edu/~chenlong/226/Ch1Space.pdf
given to Sobolev spaces satisfying certain zero boundary conditions. Distributions and weak derivatives. We begin with the nice function space C1 0 (). Recall that it denotes the space of infinitely differentiable functions with compact support in . Obviously C1 0 is a real vector space and can be turned into a topological vector space by a ...
https://mathoverflow.net/questions/237636/are-compactly-supported-continuous-functions-dense-in-the-continuous-functions-o
Continuous functions on $\mathbb R^d$ such that the support is a compact subset of $\overline{\Omega}$? For "nice" $\Omega$ this would be the space of continuous functions on $\Omega$ vanishing at the boundary. $\endgroup$ – Jochen Wengenroth Apr 29 '16 at 12:50
https://en.wikipedia.org/wiki/Sobolev_spaces_for_planar_domains
Support properties: Let Ω c be the complement of Ω and define restricted Sobolev spaces analogously for Ω c. Both sets of spaces have a natural pairing with C ∞ (T 2). The Sobolev space for Ω is the annihilator in the Sobolev space for T 2 of C ∞ c (Ω c) and that for Ω c is the annihilator of C ∞ c (Ω).
https://www.encyclopediaofmath.org/index.php/Function_of_compact_support
The support of is the closure of the set of points for which is different from zero . Thus one can also say that a function of compact support in is a function defined on such that its support is a closed bounded set located at a distance from the boundary of by a number greater than , where is sufficiently small.
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