Sobolev spaces are function spaces generalising the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, functions in a Sobolev space have also differentiability conditions: that is, we require all partial derivatives of the function up to a ...
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29 views
What is the Sobolev Lemma?
In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult.
The ...
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35 views
Showing that smoothing operators are compact
Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator?
I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
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1answer
27 views
finite elements-exercice
We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$
Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
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1answer
17 views
$g=(g_1,…,g_N)$ $Q$ periodic implies $\int_Q \operatorname{div} g=0$?
Let $g_i:\mathbb{R}^N\to\mathbb{R}$ ($i=1,...N$) with $g_i\in W^{1,\infty}(\mathbb{R}^N)$ and define $g=(g_1,...,g_N)$. Let $G=\operatorname{div}g$, where $\operatorname{div}g=\frac{\partial ...
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24 views
Simple heat equation, solution regularity
I have a small problem with a regularity result for a simple parabolic heat equation:
Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
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1answer
20 views
Mean value theorem in sobolov space under integral
Sorry this seems like a basic question, but I'm having trouble figuring out the answer.
Let g(x) be the step function over [-1,1] and f(x) a function with $f\in H^1[-1,1]$, that is it has a square ...
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36 views
about a proof in chap 5 in PDE evans book
in the proof of the theorem 5 page 281 of Evans ( PDE - Evans ) , he write the Morrey estimate. he writes in the estimate : $y \in B(x,r)$
After in the proof we have
$$ |v(y) - v(x)| = |u(y) - ...
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1answer
73 views
Fourier transform in $L^2$
I have a function $\phi\in L^2(\mathbb{R}^3)$ and I know that its Fourier transform satisfies the following equation:
$$(p^2-A)\hat{\phi}(p)=Q\frac{A+\lambda}{p^2+\lambda}$$
where $Q$ is a constant, ...
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1answer
36 views
Elimination of a singularity
Let $z=a+ib$ with $b>0$; the function
$$f(x)=\frac{e^{iz|x|}}{|x|}$$ is in $L^2(\mathbb{R}^3)$; in fact
...
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1answer
47 views
Sum of Banach spaces
Let $H^2(\mathbb{R}^3)$ the usual Sobolev space and consider the following set
$$X=\bigg\{u\bigg|u=\phi+\frac{Q}{|x|},\phi\in H^2,\,\, Q\in\mathbb{C}\bigg\}$$
I observe that the decomposition is ...
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0answers
47 views
Supremum of norms of line integrals
I have the following problem:
Let's say $\Omega\subset\mathbb{R}^2$ is a bounded open connected set with Lipschitz boundary and $f\colon\Omega\rightarrow\mathbb{R}$ is a function such that $f\in ...
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0answers
51 views
Almost every restriction is absolutely continuous
I'd like to prove the following: let $B_r(x)$ be the open disk of centre $x$ and radius $r$ contained in $\mathbb{R}^2$, and let $f \in H^1(B_1(0))$ ($H^1 = W^{1,2}$). Fix $\rho < 1$.
Then, for ...
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0answers
31 views
Some ideas about $H=W$
Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ .
Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$
where ...
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0answers
53 views
Lagrange theorem
I have this example and I don't understand its resolution:
Let $\Omega \subset \mathbb{R}^n , n\geq3$ be a bounded open set (with smooth boundary), let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ be ...
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1answer
32 views
$W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$?
How can I prove that $W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$ if $s_1 > s_2 + n/4$ ? $W^{s,p}$ denotes a general Sobolev space for $s =0,1,2,\cdots$. The hook means a ...
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0answers
82 views
Integrating $\ln(1+|\ln|x||)$ in $B_1(0)$
I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$
Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and ...
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0answers
55 views
Compact embedding theorem of $W^{k,p}(R^n)$?
Is there some kind of function space $X(R^n)$ which satisfies the compact embedding relation as follows: $W^{k,p}(R^n)\hookrightarrow\hookrightarrow X(R^n)$? Could I guess the indeterminate function ...
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0answers
58 views
Good books on the relationship between function spaces $C^m([0,T],H^k(\Omega))$ and $C^r([0,T]\times\Omega)$ in details?
Good books on the relationship between function spaces $C^m([0,T],H^k(\Omega))$ and $C^r([0,T]\times\Omega)$ in details? I realy want to learn some knowledge on this kind of function spaces, but I ...
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votes
0answers
47 views
Time dependent or Bochner space references?
Does anyone have any recommendations where I can learn about time dependent or Bochner spaces? I mean spaces like $L^p(0,T; H^{-1}(\Omega))$. I think one needs some knowledge of distributions, so any ...
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0answers
47 views
Proving $ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) $ by using the Sobolev inequality.
Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ...
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0answers
29 views
Conditions on $p(x)$ and $q(x)$ in energy scalar product
What are the most general conditions on $p(x)$ and $q(x)$ so that the following
is a scalar product (this is sometimes called the energy scalar product):
$$
\langle f\mid g\rangle = \int_0^\infty ...
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votes
0answers
29 views
About boundness of a composite function.
Let $u = u(t,x) \in C^0 ([0,\infty), W^{m,2} ) \cap C^1 ([0,\infty), W^{m-1,2} ) \cap C^\infty ([0,\infty) \times \Bbb R^n)$ for $m = 0 , 1 , \cdots . , $ and $f \in C^\infty$.
Then I know the fact ...
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0answers
27 views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega ...
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0answers
45 views
Notational question
$\def\Mat{\mathop{\mathrm{Mat}}\nolimits}\def\R{\mathbb R}\def\tr{\mathop{\mathrm{tr}}}\def\abs#1{\left|#1\right|}\def\norm#1{\left\|#1\right\|}$
In the map $ f'\colon \Mat_n(\R) \to \Mat_n(\R)^*$ ...
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0answers
50 views
Is this family of projections $\nu\mapsto P_\nu$ Lipschitz continuous?
For $\nu\in (\epsilon,1)$ with $0<\epsilon<1$, let $P_\nu:H_0^1(\Omega)\rightarrow H_0^1(\Omega)$ with $\Omega\subset \mathbb{R}^N$ bounded Lipschitz domain, be the projection operator onto the ...
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0answers
29 views
Is true that $C^{0,\alpha}(\Omega) \cap H^{1}_{\phi}(\Omega)$ is dense in $W^{1}_{\phi}(\Omega)$?
Let $\Omega$ be a domain bounded smooth and $\phi \in H^{1}(\Omega)$. Denote by $$W^{1}_{\phi}(\Omega) = \{u \in H^{1}(\Omega): u = \phi \ \ \mbox{on} \ \partial \Omega \ \ \mbox{ in the sense of the ...
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0answers
108 views
How can we glue Sobolev functions?
Let $u:A\cup B\to R$ be a function (where $A$ and $B$ are disjoint sets and $A\cup B$ is connected) such that $u$ restrict to $A$ and to $B$ are in $W^{1, p}$. Which result guarantees me that $u$ is ...
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0answers
108 views
Reference of Sobolev space
currently I'm using Krylov's book, while consulting Evans (too many details are left out, for my level).
Also, Adams 1975 version has been widely cited.
So besides these ones, which book in your ...
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1answer
90 views
What is “approximation in a Sobolev Space”? For example,
I want to know the meaning of the statement as below.
$$ \text{There is a sequence} \;\{f_k\} \subset W^{3,2}\; \text{approximating}\;\; f \;\;\text{in}\;\; W^{2,2}. $$
Here $ W^{n,m} $ means a ...
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votes
1answer
60 views
When Dirac function is in $H^{-m}(R^n)$?
If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
-1
votes
1answer
57 views
There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?
There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?
Justify your answer!
Thanks.
-1
votes
2answers
46 views
When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?
Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth.
Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
-1
votes
1answer
52 views
Show reflexivity of Sobolevspace $W^{1,4}(0,1)$
I would like to show elementary - using the canonical embedding - that the Sobolevspace $W^{1,4}(0,1)$ is reflexive.
Therefore I set $X=W^{1,4}(0,1)$ and now I have to show that the canonical ...
-1
votes
1answer
27 views
if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then$ f\in H^k (\Omega')$
prove that if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then $f\in H^k (\Omega')$
-1
votes
1answer
24 views
An inequality : $ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$
Let $u=u(x)$ be a real-valued function defined on $\mathbb R$.
How does this inequality hold?
$$ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$$
...
