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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Boundary equality
Le $\Omega$ be a bounded domain with Lipschitzian boundary. How can I show that $$\left\langle\frac{\partial u}{\partial n},h\right\rangle_{W^{\frac{1}{p' },p}}=0, \ \forall h\in W^{1,p}\implies \frac{...
0
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0answers
17 views
Maximal monotone functional
Given a bounded domain $\Omega$ $h:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by
$$ h(u)=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\mathbb{R}^N}dz$$ why is it continuous and maximal monotone ?
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1answer
40 views
Does derivative imply weak derivative?
It is known that there are functions whose weak derivative exists but (classical) derivative does not exist. I want to confirm that "any differentiable function is weakly differentiable". Help me.
-1
votes
1answer
30 views
Is $C^k$ function space is dense in Sobolev space?
It is known that $C^\infty$ function space dense in Sobolev space. But I want to know that is this statement true also for $C^k$ function space?
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1answer
23 views
Sequentially continuity
Given a map $E:L^\infty\rightarrow C^1(\overline{\Omega}), g_n\rightarrow g$ in $L^\infty(\Omega)$. $u_n=E(g_n), n\in \mathbb{N}$ and u=E(g) and show that every time we take a subsection we can find ...
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0answers
22 views
Inequality involving periodic functions and Sobolev space.
Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if
$$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 ...
1
vote
0answers
35 views
Evans pde proof of global approximation theorem
I have a question regarding step 3 of the proof (shown in picture below).
I am wondering if there is a need for us to consider the subset $V\Subset U$ instead of directly using $U$ when showing the ...
0
votes
1answer
28 views
Sequentially lower semicontinuity
Let $\Omega$ be a bounded open set with lipschitz boundary, How can we show that the functional defined by $f:W^{1,p}\rightarrow\mathbb{R}$ $f(u)=\int_{\Omega}|Du|_{\mathbb{R}^N}^p$
is sequentialy ...
1
vote
0answers
22 views
Reference request: Density of testfunctions in sobolev space $ W^{1}_{0}$
can someone help me out and name a good source where the following statement ist proven?
$C^\infty_0(\Omega)$ is dense in $W^{1,p}_0(\Omega)$ with $\Omega \subset \mathbb{R}^n$ being an open, bounded ...
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0answers
9 views
well-posedness of elliptic pdes with gaussian weights
Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by
$$
\gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}.
$$
Let $d\gamma_n$ denote the following measure ...
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2answers
55 views
Convergence of sequence of function for a bounded sequence in the Sobolev space
Let $u_n$ be a bounded sequence in $W_{0}^{1,p}(\Omega)$. Then upto a subsequence one has
$$
u_n\to u \mbox{ weakly in}\,W_{0}^{1,p}(\Omega).
$$
How the following statement is true?
$$
\int_{\Omega}|\...
1
vote
1answer
62 views
Sobolev Embedding into $L^{\infty}$
I tried to find the question here, but I couldn't.
I'm a bit puzzled by Sobolev embeddings at the moment.
In my lecture notes, I found the statement
"If $\Omega \subseteq \mathbb{R}^d$ is ...
0
votes
1answer
19 views
Measurability of $\nabla \cdot (a \cdot \nabla u)$ implies measurability of $u$?
Suppose I know that $\nabla \cdot (a(t)\nabla u(t))$ is such that $\nabla \cdot (a\nabla u) \in L^2(0,T;L^2(\Omega))$ on some bounded domain $\Omega$. Here $a\colon [0,T] \to \Omega$ is such that $a \...
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votes
1answer
54 views
Fractional Sobolev norm of a radial function
Let $f(x):\mathbb{R}\to \mathbb{R}$ be a smooth function, with $\operatorname{supp}(f) \subset [0,1]$.
Consider the radial function $F:\mathbb{R}^d\to \mathbb{R}$, defined by $F(x) := f(|x|)$.
A ...
1
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0answers
30 views
How to show this Sobolev inequality?
I am currently studying numerical analysis and stumbled upon the following task:
Prove that for any $q \in [2, +\infty)$, there exists $C \gt 0$ such that
$\Vert f\Vert_{L^{q}(\mathbb{R}^2)} \le C\...
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votes
1answer
14 views
Weak Convergence of sequence in a Sobolov Space.
Consider the question asked in here.
I understood most of the answer in the question but the part about the weak convergence I did not get. To show that $u'_{n_k}$ converges weakly to $u$ in $L^p$, ...
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0answers
20 views
Norm Equivalence in Sobolev Space
I am attempting to demonstrate that for all $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$
\begin{align*}
\int_U |u|^p \ dx \leq C\int_U |Du|^p \ dx
\end{align*}
where $U$ is some open subset in $\mathbb{R}^n$....
5
votes
0answers
58 views
How is “continuous dependence on initial conditions” defined? And how to prove it?
EDIT:
I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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0answers
18 views
Equivalence of Gradient and Hessian Norms in Sobolev Space
I am attempting to demonstrate that for $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$
\begin{align*}
\int_U |Du|^p \ dx \leq C\int_U |D^2u|^p \ dx
\end{align*}
where $U$ is some open subset of $\mathbb{R}^n$.
...
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votes
0answers
18 views
understanding Hilbert Sobolev spaces
We denote the space $\dot{H}^k$ for the homogeneous Sobolev space and $H^k$ for the inhomogeneous Sobolev space where $H^k=W^{k,2}$. It is true that de dual of $\dot{H}^k$ can be identified with $H^k$...
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1answer
25 views
Sobolev imbedding theorem $H^{1,p}(\mathbb{R}^n)$ contained in $L^{{np}/(n-p)}(\mathbb{R}^n)$ (Taylor Michael)
i. I can not make sense of the following:
For (2.4) I imagine that it is the fundamental theorem of the calculation but I can not prove it formally.
Neither will it be understood how to arrive at ...
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0answers
15 views
Using Hille-Yosida to show existence of solution to differential equation.
Suppose $I=(0,l)$ is an interval and $v\in C(\overline{I})$. Consider the differential equation
$$\begin{cases}u_t=u_{xx}+v(x)u_x,\\ u(0,t)=u(l,t)=0,\;\;\\u(x,0)=u_0.\end{cases}$$
where $u=u(x,t):I\...
1
vote
0answers
8 views
Sobolev spaces on domains and manifolds, what is the difference?
What are the (technical) differences between Sobolev spaces on domains $\Omega \subset \mathbb R^n$ or (compact) manifolds such as two-dimensional spheres?
1
vote
1answer
25 views
Are 2 Hilbertspaces with different inner products identical if the associated norms are equivalent?
suppose we have the Sobolev space $H^1_0(\Omega)$ over a bounded domain $\Omega \subset \mathbb{R}^2$. With the standard inner product it sure is a Hilbert space. BUT: What if we equip $H^1_0$ with ...
1
vote
0answers
23 views
Schauder regularity
My question is: If $\Omega$ is bounded with smooth boundary...
Let $f \in C^{0,\alpha}(\overline{\Omega})$, and let $u$ be a weak solution of the Poisson equation $-\Delta u = f$. Then $u \in C^{2,\...
3
votes
1answer
49 views
$f(x)=\sin(x)\in H_{0}^{1}(\Omega)$ with $\Omega=(0,\pi)$?
We know that $f\in C^{\infty}(0,\pi)$ and $f(0)=f(\pi)=0$. But $supp\ f=[0,\pi]$. From the definition, $H_{0}^{1}(0,\pi)$ is the clusear of $C_{0}^{\infty}(0,\pi)$ in $H^{1}(0,\pi)$. Since, $f$ is not ...
3
votes
1answer
64 views
We could identity $H^{-1}$ with $H_0^1$ but we don't. Why?
Today in our lecture on partial differential equations while discussing dual spaces of Sobolev spaces:
We could identify $H^{-1}$ with $H_0^{-1}$ by the $H_0^1$ inner product (Riesz) But won't and ...
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0answers
32 views
How to understand the definition of $W_0^{1,2}(\Omega)$
We defined $W_0^{1,2}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$.If $u \in W_0^{1,2}(\Omega)$ then u is compactly supported , but is it also right for |u| ?
3
votes
0answers
49 views
Density of smooth functions in the weighted Sobolev space $W^{1,p}((0,1),w)$, where $w(x)=x^{\lambda}$, $\lambda>0$
Let $1<p<\infty$ and $w(x):=x^{\lambda}$, where $\lambda>0$. One way to define the weighted Sobolev space $W^{1,p}((0,1),w)$ is the following (see [1]): we say that $u\in W^{1,p}((0,1),w)$ if ...
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vote
1answer
43 views
Why is $f_\epsilon(u) \in W_0^{1,2}(\Omega)$?
For $\epsilon>0$ let $f_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$
One calculates that $\nabla f_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u $ , for $\epsilon$ to 0 this term goes to $\...
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votes
1answer
81 views
Do invariant functions form a Banach (sub)manifold in function spaces?
Let $G$ be a topological group, and $X$ some function space; preferably a Sobolev space $X=W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is some invariant subset ($g\Omega \subset \Omega$) or ...
1
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0answers
40 views
$u \in W_0^{1,2}(\Omega) \Rightarrow |u| \in W_0^{1,2}(\Omega)$
For $\epsilon>0 $ define $g_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$.
One finds $ \nabla g_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u$
and $ g_\epsilon(u)\in W_0^{1,2}(\Omega)$ .
Then ...
2
votes
1answer
36 views
Deriving Variational Formulation
Determine the variational formulation of
\begin{cases}
-\Delta u+u=xy \quad& \text{in } \Omega\\
\nabla u\cdot \vec{n}+2u=3 \quad& \text{in } \partial\Omega
\end{cases}
What I have tried:
\...
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votes
0answers
38 views
Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?
We know that when $I$ is a bounded interval and $1<p\leq \infty$ that the injection $W^{1,p}\subset C(\overline{I})$ is compact.
The proof of this fact uses the Arzela-Ascoli theorem on the unit ...
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votes
1answer
28 views
Is every linear finite element space over a bounded domain a subspace of the sobolev space H^1?
Since my knowledge of functional analysis, $L^p$-, Sobolev- and Hilbert spaces is not very good, I thought I could ask...
Suppose we have a domain $\Omega \subset \mathbb{R}^2$ which is continuously ...
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votes
0answers
28 views
$||.||_{\infty}$ in sobolev space
Let $u_n ∈ W^{1,1} (I), I=(0,1)$ defined by:
$u'_n (x) = n$ if $x < 1/n$
$u'_n (x) = 0$ if $x > 1/n$
$u_n (0) = 0$.
Find $||u_n − 1||_∞$.
My attempt:
We have $u_n(x)=u_n(0)+\int_0^{1/n}ndt=...
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votes
0answers
35 views
Estimates for Poisson's equation
I am studying Calderón-Zygmund estimates ($L^p$ estimates) for Poisson equation. What I already know is:
Let $\Omega$ be a bounded domain, $f \in L^p(\Omega)$, $1<p<\infty$, and let $w$ be the ...
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vote
1answer
29 views
The dual space of the Sobolev space $W^{k,p}(\Omega)$.
Let $\Omega$ be a nice domain in $\Bbb R^n$. It is known that any element $T\in\left( W^{k,p}(\Omega)\right)^*$ admits a (possibly non-unique) representation of the form
$$
Tu = \sum_{|a|\le k} \int_\...
1
vote
0answers
22 views
Orlicz space is complete under the Luxemburg norm
I want to prove that Orlicz space is complete. Here is my attempt
Let $(X,\mu)$ be a measurable space and suppose $\phi(x)$ is a young function. Define $$L^{\phi}=\{f\hspace{0.2cm} \text{measurable:}\...
2
votes
0answers
31 views
Question on proper choice of test function
Let $\Omega$ be a $\mathcal{C}^2-$compact manifold in $\mathbb R^2$ and consider the non-homogeneous heat equation:
$$
\partial_t v - \Delta v
= f
\quad \text{in } \ \Omega \times (0,T),
$$
where $f \...
2
votes
0answers
32 views
function is convex in Sobolev space
Let $u ∈ W^{1,1} (]0, 1[)$ and
$$J_\varepsilon(u) =\frac{1}{2}\int_0^1(\varepsilon+x^\alpha)u'^2dx+\frac{1}{4}\int_0^1u^4 dx-\int_0^¹uf dx.$$
Prove $J(u)$ is convex.
My argument is sum of 3 ...
1
vote
2answers
36 views
Sobolev embedding when $p=n$: $W^{1,p}(\mathbb{R}^{n}) \hookrightarrow L^{q}(\mathbb{R}^{n})$ for $q: p \leqslant q < \infty$
In class, aside from the standard Gagliardo-Nirenberg-Sobolev and Morrey inequalities, my professor also covered the case when $p=n$. In particular, if $p=n$, then $W^{1,p}(\mathbb{R}^{n}) \...
1
vote
1answer
38 views
Compact embedding of fractional space
Is the space $H^\lambda((0,T); H^1(K))$ for $0 <\lambda <1$ where $K$ is compact subset of $\mathbb{R}^n$ compactly embedded in $L^2( (0,T) \times K)$?
$H^\lambda((0,T); H^1(K))\hookrightarrow \...
1
vote
1answer
52 views
Question about the Sobolev space $W^{1,p}(I)$.
Let $\{u_n\}$ be a bounded sequence in $W^{1,p}(I)$ where $1<p\leq\infty$ and $I$ is bounded. Then
there is a subsequence $\{u_{n_k}\}$ and $u\in W^{1,p}$ such that $\|u_{n_k}-u\|_{L^\infty}\to
0$ ...
2
votes
0answers
32 views
Interpolation of weak derivatives
Is it true that if a function $f \in L^p$ has the property that for all $\vert\alpha\vert=k$ we have $D^{\alpha} f \in L^p$ then $D^{\alpha} f \in L^p$ for all $\vert \alpha \vert \le k$? This is ...
1
vote
1answer
35 views
$u \in H_0^1(\Omega) \implies u^+ \in H_0^1(\Omega)$ (boundary not $C^1$)
I know how to prove that $u \in H^1(\Omega) \implies u^+ \in H^1(\Omega)$ since this is Exercise 5.18 in Evan's PDE book. However, I'm not sure how to extend this to $H_0^1(\Omega)$. My first ...
1
vote
0answers
20 views
Are the derivatives of the orthogonal polar factor locally integrable?
Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a real-analytic map, satisfying $\det df>0$ everywhere except on a set of Hausdorff dimension not greater ...
0
votes
0answers
29 views
Is it true that $\int_{\Omega} u^2\leq \alpha \int_{\Omega} |Du|^2$?
Let $u\in H^1_0(\Omega)$, where $\Omega$ is a bounded open set in $\Bbb{R}^2$. Is the following true:
For some $\alpha>0$, $\int_{\Omega} u^2\leq \alpha \int_{\Omega} |Du|^2$
I remember reading ...
8
votes
0answers
130 views
Is the normalized derivative of a holomorphic function Sobolev?
Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic.
More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
0
votes
0answers
16 views
Lipschitz constant in a bounded open set and infinity sobolev derivative
Let $V$ be a bounded open set. Let $f:V\rightarrow\mathbb{R}$ such that $f\in C(\overline{V})$ and $f$ is Lipschitz function on V ($Lip(f,V)<+\infty$).
I would like to show that $Lip(f,\overline{V})...
