1
vote
1answer
35 views

Completeness of Sobolev space constructed from seminorm

Define $W^{p,r}(\mathbb{R}^d):=\{f\in L^p(\mathbb{R}^d) : D^\alpha f\in L^p(\mathbb{R}^d), \forall 0<|\alpha|\le r\}$ where $1\le p\le\infty$. Let the seminorm on $W^{p,r}(\mathbb{R}^d)$ be ...
3
votes
1answer
106 views

Equivalent norms on a cartesian product of Hilbert spaces.

Notation: $H_0^1=H_0^1(a,b)$, where $-\infty<a<b<\infty$. Let $\|\cdot\|_V$ be a norm on $V:=H_0^1\times H^1_0\times H^1_0$ given by ...
2
votes
2answers
83 views

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...
2
votes
1answer
64 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
2
votes
1answer
72 views

$C_0^\infty(0,T)\cdot V$ dense in the Bochner space $L^2(0,T;V)$

Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and ...
2
votes
1answer
55 views

How to show this Sobolev space is a uniformly convex space?

From the book by Kufner: How do I prove this theorem? I'd like to do it using the epsilon delta definition (see http://en.wikipedia.org/wiki/Uniformly_convex_space) if possible.
3
votes
0answers
97 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
0
votes
1answer
88 views

Sum of Banach spaces

Let $H^2(\mathbb{R}^3)$ the usual Sobolev space and consider the following set $$X=\bigg\{u\bigg|u=\phi+\frac{Q}{|x|},\phi\in H^2,\,\, Q\in\mathbb{C}\bigg\}$$ I observe that the decomposition is ...
2
votes
1answer
137 views

Is this set dense in $H^1(\Omega)?$

Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$ dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$ Here $f$ is some linear functional so that $V_1$ is also ...
3
votes
1answer
97 views

Is $W_0^{1,p}(\Omega)\cap L^q(\Omega)$ Uniformly Convex?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $p<q$ with $p,q\in (1,\infty)$. Is $W_0^{1,p}(\Omega)\cap L^q(\Omega)$ Uniformly Convex with respect to the norm: ...
3
votes
1answer
70 views

Is $W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$ complete?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. Define $W=W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$. Is $W$ complete with respect to the norm $\|v\|=\|v\|_{1,p}+\|v\|_\infty$? If $u_n$ is a ...
3
votes
1answer
88 views

Extending definition of weighted $L^2$ norm

Is there a simple characterization of the domain of the semi-norm $\| \nabla (g \ast x) \|_{L^2 ( \mathbb{R}^3)}$, where $g$ is a gaussian convolution kernel? It is finite on $L^2$, but probably on a ...
3
votes
1answer
260 views

Space Sobolev $W^{m,p}$ complete

Show that Sobolev space is complete. I am trying Than $L^p(\Omega)$ is complete then If $f_n \in L^p(\Omega)$ then $f_n \to f \in L^p(\Omega)$. But rest show that $D^{\alpha}f \in L^p(\Omega)$. How I ...
3
votes
3answers
263 views

Question about Sobolev embedding theorem

The Sobolev embedding theorem as stated in my notes says that if we have $k > l + d/2$ then we can continuously extend the inclusion $C^\infty(\mathbb T^d) \hookrightarrow C^l(\mathbb T^d)$ to ...
4
votes
1answer
88 views

The reflexivity of the product $L^p(I)\times L^p(I)$

I am starting to read about the Sobolev spaces $W^{1,p}(I),$ where $I$ is an open interval in $\mathbb{R}.$ In order to establish the reflexivity of $W^{1,p}(I)$ for $p\in ]1,\infty[,$ I need the ...
3
votes
1answer
82 views

comparison between spaces

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of ...
3
votes
2answers
364 views

Approximate a positive Sobolev function by positive smooth functions

Here is a problem that I have encountered in PDE book several times. But I have never seen a proof of it. I will be very grateful if someone could give me a proof. Question: Let $B$ be the unit ball ...
7
votes
1answer
944 views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?