Questions tagged [sobolev-spaces]
For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.
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0answers
11 views
Question on the dimension of the boundary of a bounded domain
Motivated by this post Dimension of boundary of a bounded domain; what to use for Sobolev inequalities and since the provided answer didn't help me that much, I thought I should just ask once more.
...
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0answers
21 views
Approximation of a $W^{1,2}(\Omega,\mathbb{S}^2)$ function
Suppose $\Omega \subset \mathbb{R}^3$ is bounded domain with smooth boundary and $u \in W^{1,2}(\Omega,\mathbb{S}^2)$ such that $|u|=1$ a.e. in $\Omega$. It is known that there is a sequence $u_n \in ...
1
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1answer
25 views
Is continuous increasing function in $H^1([0,1])$
Consider a function $f(x):[0,1]\rightarrow \mathbb{R}$. If $f(x)$ is continuous and increasing, is $f(x)$ in $H^1(\Omega)$, the Sobolev space with norm $\sqrt{\int_0^1 (|f(x)|^2 + |Df(x)|^2) dx}$?
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0answers
19 views
Understanding iterated covariant derivatives to define Sobolev spaces on manifolds
I'm having big troubles understanding the definition of Sobolev spaces on manifolds.
Ok, so we have a Riemannian manifold $(M, g)$, and then we can define a natural riemannian measure (which I will ...
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2answers
41 views
Does strong convergence in $H$ (or $L^2$) imply convergence in $V$ (or $W_0^{1,2}$)?
Suppose:
$\mathbf{V}$ and $\mathbf{H}$ are Hilbert spaces.
$\mathbf{V} \hookrightarrow \mathbf{H}$ is compact embedding.
$\mathbf{V}$ is dense in $\mathbf{H}$.
For example $\mathbf{V} = W_0^{1,2}(\...
1
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0answers
22 views
Integrability of the Fourier transform in Sobolev space
I believe that the statement below is a standard fact but I haven't figured out yet:
Suppose $f\in L^{1}(\mathbb{R}^{n})$ has integrable partial derivatives of order $n+1$ and $D^{\alpha}f\in L^{1}(...
1
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0answers
8 views
Trace of $H^1(\Omega)$ function onto regular hypersurface inside $\Omega$ (and not the boundary)?
Let $\Omega$ be a smooth domain in 2D and let $S$ be a closed smooth surve inside $\Omega$
Do functions in $H^1(\Omega)$ have a trace on $S$ and does it follow that $u \in H^1(\Omega) \implies u|_{S}...
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0answers
31 views
Can Analytic Functions on the Circle be Characterized by Sobolev Norms?
In Exercise I.4.4 of Katznelson's book An Introduction to Harmonic Analysis we learn that the Fourier series $$f(z) = \sum_{k=-\infty}^{+\infty} f_k z^k$$ defines an analytic function in the ...
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0answers
25 views
Weak convergence of weak derivative implies strong convergence
Note that here both $w_n$ and $\phi_n$ are vector functions.
The following is given :
$ w_n \rightarrow w$ weakly in $ L^q(B; {\mathbb{R}}^M)$
& $\{curl[w]\}_{n=1}^{\infty}$ is precompact in $ ...
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1answer
16 views
Approximating Sobolev functions in $W^{1,p}(\mathbb{R}_+^n)$
Let $p \geq 1$. I know that there exists a continuous and linear extension operator $$ E : W^{1,p}(\mathbb{R}_+^n) \to W^{1,p}(\mathbb{R}^n) .$$
I read that from the existence of such an extension ...
3
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1answer
77 views
Estimate of a weak solution in a nonhomogeneous equation
$\textbf{Problem}$ Let $\Omega \subset \mathbb{R}^n$ be open, bounded and connected with $\partial \Omega\in C^1$. For each $i,j=1,\cdots,n$, assume that $a_{ij},b_i,c \in L^{\infty}(\Omega)$ (real ...
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0answers
24 views
Existence of minimizers in Evans chapter 8
I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ ...
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2answers
20 views
Why is $L³$ continuously embedded into $H^{-1}$, the dual of $H¹_0$?
Why is $L³$ continuously embedded into $H^{-1}$, the dual of $H_0^1$?
In this article https://drive.google.com/open?id=0B0MDtuRPAebZQ3NvRFVQekJ4VEpTMldiRWxmbU9GLVM4MDNF the authors claim, in the ...
2
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1answer
29 views
Closed subset of $W^{1,2}([a, b], \mathbb{R})$
In the context of weak solutions of boundary value problems, I want to show that the set $$\{u \in W^{1, 2}([a, b], \mathbb{R}) \; : \; u(a) = 0 = u(b) \}$$ is closed in $W^{1,2}([a, b], \mathbb{R})$. ...
3
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1answer
50 views
Bound gradient in $H^2_0(\Omega)$ by Laplacian
Let $\Omega\subseteq \mathbb{R}^n$ be an open set and show that
$$
\lVert{\nabla u\rVert}_{L^2}^2 \leq \epsilon\lVert{\Delta u\rVert}_{L_2}^2 + \frac{1}{4\epsilon}\lVert{u\rVert}_{L^2}^2
$$
for any $\...
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0answers
22 views
Example 3 - Section 5.5.2 - Partial Differential Equations by Evans (2)
Could somebody help me: there is example from book:Example
And I don't understand why:
$
u_{x_i}=-\frac{\alpha x_i}{|x|^{\alpha+2}}
$.
I thought that $|x|=|x_1+x_2+...+x_n|$. And here is my ...
0
votes
1answer
25 views
prove $\rho\in K\to ||\sqrt{\rho}'||_{L^2}^2$ is convex, where $K$ is a convex cone
$$K = \{\rho\in H^1(0,1);\rho \ge 0,\sqrt{\rho}\in H^1(0,1)\}$$
I have proved that $K$ is a convex cone. Now I'm asked to prove that $\rho\in K\to ||\sqrt{\rho}'||_{L^2}^2$ is convex.
I tried ...
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1answer
30 views
Continuous but not compact Sobolev embedding
Let $U \subset \mathbb{R}^n$ be an open ball and $p^* = \frac{pn}{n-p}$ be the Sobolev conjugate. Show that, for $p=2$, $W^{1,p}(U)$ cannot be compactly embedded in $L^{p^*}(U)$.
Now, from the ...
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2answers
70 views
Why weak convergence and a.e. convergence imply the convergence of this integral?
In the proof of the Brezis-Nirenberg theorem on the book "Nonlinear Analysis and Semilinear Elliptic Problems" by Ambrosetti and Malchiodi, it is used lemma 11.11 at pag. 183, in whose proof we have a ...
2
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0answers
44 views
Distributional second-order derivatives of $\frac{e^{-|x|}}{4\pi |x|}$ to show the solution of $u -\Delta u=f$ is in $H^2$
In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)...
2
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0answers
34 views
Compactly supported functions are dense in local Sobolev spaces
Let $\Omega\subset\mathbb{R}^n$ be an open set. I would like to know why $\mathcal{D}(\Omega,\mathbb{C}^l)=C^{\infty}_{\mathrm{c}}(\Omega,\mathbb{C}^l)$ is dense in $H^s_{\mathrm{loc}}(\Omega,\mathbb{...
3
votes
0answers
31 views
How to show that the embedding is compact.
If $U\subseteq \mathbb{R}^n$ is a "nice" bounded domain, then we know that the embedding $H_0^1(U) \hookrightarrow L^2(U)$ is compact.
I want to show that the the embedding $H_0^1(U) \hookrightarrow ...
0
votes
0answers
25 views
Weak derivative of radial function
I want to prove that some radial function u is weakly differentiable in R^n, how do I translate that to a condition over the real function f(r) =u(r), for r>0?
2
votes
0answers
12 views
Composition with Lipschitz map is Lipschitz on Sobolev spaces
Suppose that $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is Lipschitz with some constant $L$ and that $F(0)=0$. Then it is clear that $F$ defines a Lipschitz continuous map $L^2(\mathbb{R}^d) \...
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1answer
25 views
Theorem in Adams Sobolev spaces book requires $u\in L^p(\Omega) \cap L^r(\Omega)$ but we only have $u \in C^\infty$ so how can theorem be applied?
Let $\Omega$ be a (open) domain in $\mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1\...
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1answer
38 views
completion of $C^\infty_0(D)$ w.r.t $\|\cdot\|_\nabla$
Let $D$ be an unbounded domain in $\mathbb{R}^n$. Consider the set $C^\infty_c(D)$ with two different norms: $\|\cdot\|_\nabla$ and $\|\cdot\|_\nabla + \|\cdot\|_{L^2}$.
It is known that when $D$ is ...
2
votes
1answer
45 views
Is there a bounded domain on which Poincaré's inequality does not hold?
Suppose that $U$ is a bounded domain in $\mathbb{R}^n$. Poincaré's inequality states that (for $U$ sufficiently "nice") there exists a constant $C>0$ such that if $u\in H^1(U)$ satisfies $\int_U u =...
1
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1answer
19 views
an upper bound of a $d$ dimensional vector under a length constraint.
Let $k = (k_1, \cdots, k_d) \in \mathbb{N}^d$ and $s = (s_1, \cdots, s_d) \in \mathbb{R}_+^d$. define
$$k^s = (k_1^{s_1},\cdots,k_d^{s_d})\in \mathbb{R}_+^d.$$
Let $|\cdot|_p$ denote the $p$-norm on ...
2
votes
1answer
34 views
Riesz representation and inverse operator.
In class, my professor went through the following construction: Let $\Omega$ be a bounded domain and define $X$ to be $H_0^1(\Omega)$ or $H^1(\Omega)$. We also define $A: X \to X^\prime$ (where $X^\...
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1answer
29 views
Sobolev spaces for vector-valued functions
Let $ \Omega \subseteq \mathbb{R}^3$ be open. How is defined the space $W^{1,p}(\Omega)$ for vector valued functions $f:\Omega \to \mathbb{R}^3 $? What is the norm in $W^{1,p}(\Omega)$?
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0answers
19 views
How to prove that a linear differential operator generates a semigroup?
I am working on a nonlinear PDE and I should now prove that the linear operators $L:=\partial_x^3+\partial_x^2$, $S(u):=u L$ and $T(u):=L+S(u)$ are generators of a $C_0$ semigroup such that $\|e^{-s A}...
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0answers
36 views
Clarification for a proof about Lipschitz approximation in $W^{1, p}(\mathbb{R}^n)$
I was reading a proof about the Lipschitz approximation of functions $u\in W^{1, p}(\mathbb{R}^n)$. There the author defines a set
$$E_{\lambda}=\{x\in\mathbb{R}^n:M|\nabla u|(x)\leq \lambda\}, \quad ...
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0answers
61 views
Order of convergence for spline interpolation in a Sobolev norm
We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{\text{loc}}(\mathbb{R})$.
Let us be more precise:
Let $h\in \mathbb{R}_{>...
1
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1answer
43 views
A definite integral inequality
Suppose $f(x)$ has continuous derivative on $[-\pi, \pi]$, $\,f(-\pi)=f(\pi)\,$ and $\,\int_{-\pi}^{\pi}\, f(x)\, dx=0$. Then prove that:
$$
\int_{-\pi}^{\pi} [\,f'(x)]^2\, dx \ge \int_{-\pi}^{\pi} f^...
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votes
0answers
18 views
Unable to determine Sobolev space
I have the following function:
$T ( \textbf{r}) = \frac{q}{4 \pi k} \int_{s_i=0}^{L_i} \frac{1}{\left| \textbf{r}-\textbf{r}'\right|} ds_i = \frac{q}{4 \pi k} \log\left( \frac{\tan(\theta_2/2)}{\tan(\...
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0answers
25 views
$H^1(\Omega')$ is in genral not a subspace of $H^1 (\Omega)$ for bounded domains $\Omega' \subset \Omega$
Reading about Sobolev spaces I found the following statement:
$H^1 ({\Omega}')$ is not a subspace of $H^1(\Omega)$ for $\Omega'\subset \Omega$.
$\left(\text{However } \ H^1_0 (\Omega ')\subset H^1_0(...
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1answer
21 views
How to choose the Testfunction-Space?
Every time I want to transform a boundary-value-problem of the form $-u'' + ... = f$ to it's weak formulation, I have problems choosing the testfunction-space.
I have the feeling that in scripts the ...
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votes
1answer
30 views
How can a discontinuous function belong to $C_B^1(\Omega)$, the space of continuous functions $u$ with bounded derivatives?
Let $\Omega = \{(x,y) \in \mathbb{R}^2 \ : \ 0 < |x| < 1, \ 0 < y < 1\}$ and consider the function $u$ defined on $\Omega$ by (Sobolev Spaces by Adams, page 68, Example 3.10)
$$
u(x,y) =
\...
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0answers
15 views
Equivalent definition for capacity in Sobolev spaces.
Define the capacity of set $E\subset\subset\Omega$, $\Omega$ bounded, as
$$ \text{cap}_p(E, \Omega)=\inf_{u\in \mathcal{F}_p(E, \Omega)}\int_{\Omega}|\nabla u|^p, $$
where the family of functions is
$...
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0answers
15 views
Continuous embedding of weighted Lebesgue space
Let $w$ belong to the class of Muckenhoupt weight $A_p$ for some $1<p<\infty$ and define the weighted Lebsgue space $L^p(\Omega,w):=\{u:\Omega\to\mathbb{R} \text{ measurable }: ||u||=(\int_{\...
6
votes
1answer
58 views
What is the orthogonal complement of $H^1_0$ in $H^1$?
Let $\Omega$ be a closed domain with smooth boundary in $\mathbb{R}^n$. Let $H^1_0(\Omega)$ be the closure of compactly supported smooth functions under the norm $\|u\|_1 = \int_\Omega u^2 + |\nabla u|...
1
vote
1answer
51 views
Variational problem of Laplace equation
We know that:
$\Delta u=-f, \in\Omega$
$u=0$ ,on $\partial\Omega$
Is equivalent to this variational problem:
find the minimum of $J(u)$ in $C^2(\Omega)\cap C^1(\bar\Omega)$, where $J(u)=\int_\Omega[...
-2
votes
0answers
23 views
Writing explicitly the difference of two indicator-like functions (with “non-fixed domain”)
How can I rewrite explicitly the following difference of indicator functions?
$$\int_{0}^A \mathbf{1}_{\{\int_0^x f(s) \ \mathrm{d}s + y + \epsilon h> g(x) \}}(x,y,z) - \mathbf{1}_{\{ \int_0^x ...
1
vote
1answer
28 views
a question about Differential quotient operator in sobolev spaces
Define $\nabla_h u=\frac{\tau_h u-u}{h}$, where $\tau_h u=u(x_1+h,x_2,...,x_n)$.
We use $\|\cdot\|$ to denote the norm in $L^2(R^n)$.
We have the following lemma:
Lemma 1. If $u$ belongs to $H^1(...
3
votes
0answers
22 views
Weak convergence in $H^1(\mathbb R^3)$ implies convergence of integrals
Suppose that $f_n \rightharpoonup f$ in $H^1(\mathbb R^3)$ (weak convergence). Then $$\int_{\mathbb R^3} \frac{\lvert f_n(x) \rvert^2}{\lvert x \rvert} dx \stackrel{n\to \infty}{\longrightarrow} \int_{...
2
votes
0answers
31 views
prove that the space $C^{\infty}(R_{+}^n)\cap W^{1,2}(R_{+}^n)$ is dense in $W^{1,2}(R_{+}^n)$
I already know that as $C^{\infty}_c(R^n)$ is dense in $W^{1,2}(R^n)$, but I don't know why the space $C^{\infty}(R_{+}^n)\cap W^{1,2}(R_{+}^n)$ is dense in $W^{1,2}(R_{+}^n)$
Is the statement true ...
0
votes
0answers
29 views
The spaces $L^p$ with $0<p<1$
Assume that there exists a constant $ M> 0$ such that $/\hat{u}/ >M$ .
Is the following inequality true?
$\left \| \hat{u}\right \|_{L^{p}} $ $\leq $ $\left \| \hat{u}\right \|_{L^{2}} ...
1
vote
0answers
32 views
Why is the laplacian a closed operator in $W^{2,p}(\mathbb{R}^n)$?
I have read that the laplacian is a closed operator in $W^{2,p}(\Omega)$,(that is, $\Delta : W^{2,p} \to L^p$) where $\Omega$ satisfies some conditions (I need the case $\Omega = \mathbb{R}^n$ so ...
0
votes
1answer
18 views
Weak Formulation's Well-Posedness with Poincare-Friedrichs Inequality
I have arrived at the weak form for a PDE which has the following form.
Let $V=\{ v\in H^1(0,1):v(1)=0 \}$, $F$ is a bounded linear functional on V.
$$\left\{\begin{array}{ll}
\...
1
vote
1answer
48 views
Prove that $W_0^{1,p}$ is a Banach space
$\textbf{Problem}$ Prove that $W_0^{1,p}(\Omega)$ is a Banach space where $\Omega$ be an open and bounded set in $\mathbb{R}^n$
$\textbf{Proof}$ $\quad $Let $\{u_n\}$ be the Cauchy Sequence in $W_0^{...
