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Results tagged with sobolev-spaces
Search options questions only
not deleted
user 2011
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
21
votes
5
answers
17k
views
When is Sobolev space a subset of the continuous functions?
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several tim …
- 3,217
8
votes
2
answers
1k
views
Extension theory with bump function
Let $B_t(0)$ denote the $n$ dimensional ball of radius $t$ centered at the origin. Does there exist a $\phi\in C(\mathbb{R}^n)$ function with the properties:
$
\phi (x) =
\begin{cases}
1&x\in B_r(0)
…
- 3,217
4
votes
1
answer
1k
views
Sobolev-Slobodeckij spaces for p=infinity
For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,
$
\left\| …
- 3,217
3
votes
1
answer
1k
views
Bessel Potential Space inequality
The Bessel Potential Space is defined for $s\in\mathbb{R}$ as,
$H^s(\mathbb{R}^d) = \{f\in L_2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L_2(\mathbb{R}^n)\}.
$
This defines a Hilbert space …
- 3,217