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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2
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1answer
40 views

Critical Homogeneous Sobolev Embedding

For $s\in\mathbb{R}$, $1<p<\infty$, define the homogeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^{n})$ as follows: For $f\in\mathcal{S}_{0}(\mathbb{R}^{n})$ (Schwartz functions with Fourier ...
1
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1answer
71 views

Poincaré constant for a ball (circle)

I've been recently looking for a best possible Poincaré constant for a particular domains $\Omega$ (it's related to my previous question Unique weak solution to Helmholtz equation on a square) for ...
0
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1answer
29 views

Taking $\inf$ for sobolev space in different order

Let $\Omega\subset \mathbb R^N$ open bounded, smooth boundary be given. Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx+\int_\Omega(|\nabla v|^2+(1-v)^2)dx, $$ and two sets $\mathcal U:=\{u\in ...
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0answers
15 views

compact embedding in sobolev spaces ($W^{1,1}(\mathbb{R}^n)$ in $L^1(\mathbb{R}^n)$ [duplicate]

i have this question : in an example of the compact embedding, the autor gives a demonstration of : the sobolev space $W^{1,1}(\mathbb{R}^n)$ is not compactly embedded in $L^1(\mathbb{R}^n)$ and it ...
2
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1answer
21 views

Periodic Poincaré Inequality?

The classical periodic Poincaré inequality states that if $u\in H^1(\mathbb T^n)$ is such that $\displaystyle\int_{\mathbb T^n} u(x)\ dx=0$ then $$\|u\|_{L^2(\mathbb T^n)}^2\leq C_d \|\nabla ...
1
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1answer
30 views

Is there $f\in H^1(\mathbb T^n)$ such that $ \textrm{div}(f)=\sum_{j=1}^n \partial_j f=1$?

Is there any $f\in H^1(\mathbb T^n)$ such that: $$\textrm{div}(f):=\sum_{j=1}^n \partial_j f=1,$$ where $1$ stands for the constant function $x\longmapsto 1$. Thanks.
1
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1answer
37 views

Inequality for Sobolev fractional spaces

I recall that the Fourier transform of a function $f \in L^1 (\mathbb{R})$ is defined by $$\hat{f}(\xi) = \frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} f(x) e^{- i x \xi} \, dx.$$ We can define that ...
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0answers
29 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
2
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1answer
16 views

Unique weak solution to Helmholtz equation on a square

I've recently started studying the modern theory of PDEs. I studied some basic properties of Sobolev space and then started with linear elliptic PDEs. I consider the following problem: For which ...
1
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1answer
29 views

$W^{1,1}(\mathbb{R}^n)$ is not compactly embedded in $L^1(\mathbb{R}^n)$?

i have this question : in an example of the compact embedding, the autor gives a demonstration of : the sobolev space $W^{1,1}(\mathbb{R}^n)$ is not compactly embedded in $L^1(\mathbb{R}^n)$ So let ...
2
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0answers
40 views

Dense subset of Nikol'skii spaces?

I'm looking for a dense subset of the Nikol'skii space $N^{s,p}(\mathbb{R}^N)=B_{p,\infty}^s(\mathbb{R}^N)$, $s\in(0,1)$, $p\in(1,\infty)$, the definition of which is recalled below: ...
2
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1answer
47 views

Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)

For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ ...
1
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1answer
26 views

Variational formulation of elliptic mixed boundary value problem

I have some troubles with variational formulations of some pies. For instance, let's consider $\chi\in H^1(\Omega)$ as the solution to the elliptic mixed boundary value problem $$ \begin{cases} ...
2
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1answer
30 views

Extending continuous linear functional of the derivative to continuous linear functional of the function

Suppose given $f,g\in L^2(\mathbb{R})$ and $f', g' \in L^2(\mathbb{R})$, the linear functional defined by $$F(g):= \int_{\mathbb{R}} f'g' dx $$ is continuous with respect to the derivative, that is ...
0
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1answer
19 views

The norm of trace of functions in $H^\frac{1}{2}(\partial\Omega)$

Let $\textbf{A}\in(H^1(\Omega))^3$, where $\Omega\subset\mathbb{R}^3$ is a bounded convex domain with its boundary $\partial\Omega$. Now we know, on $\partial\Omega$, ...
2
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2answers
20 views

Continuous representative for functions in $W^{1,2}(\mathbb{R})$

I want to prove that $K(x,y) = \frac{1}{2}e^{-|x-y|}$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$ and as a hint I have given that for $f\in W^{1,2}(\mathbb{R})$ I should use its continuous ...
1
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1answer
23 views

Proving $0.5\exp(-|x-y|)$ is reproducing kernel for $W^{1,2}(\mathbb{R})$

Prove that $K(x,y)=0.5\exp(-|x-y|)$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$, i.e. that $K(x,y)\in W^{1,2}(\mathbb{R})$ and for the continuous representative $\hat{f}$ of $f\in ...
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0answers
32 views

Derivatives of mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
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1answer
18 views

Disproving $\|u\|_{L^1}\leq C\|Du\|_{L^1}$ for compact, smooth $u$

Show that there is no $C\gt 0$ such that $\|u\|_{L^1}\le C\|Du\|_{L^1}$ for all $u\in C_c^\infty(\mathbb{R}^2)$. Earlier I was able to prove that it is true, if you replace the norm on the left ...
4
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0answers
39 views

Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and ...
1
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1answer
28 views

The convegrence of $L^2$ norm of the gradient of a mollified sequence of functions

Let $u\in W^{1,2}(\Omega)$ where $\Omega\subset R^N$ is open bounded, smooth boundary. Given a sequence of function $u_n\in L^2(\Omega)$ such that $\nabla u_n$ is defined a.e. and $\nabla ...
0
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1answer
20 views

Show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}})$ is in Sobolev space $W^{1,p}(B_1(0))$

As part of my seminar this semester, I need to show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}}) \in W^{1,p}(B_1(0))$. I have shown that $f$ is indeed in $L^p$, but could use some help proving ...
2
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0answers
26 views

$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
2
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0answers
34 views

Multi-index notation confusion

While trying to understand a proof of equivalence of norms for $H^k(\mathbb{R}^n)$ (Fourier Transforms) I came across a possible inconsistency in the multi-index notation. Can somebody please clarify ...
1
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1answer
43 views

Weak derivative in Sobolev spaces

A function $u: \Bbb R \longrightarrow \Bbb R$ is weakly differentiable with weak derivative $v$ if there exists a function $v: \Bbb R \longrightarrow \Bbb R$ such that $$\int_{-\infty}^\infty u \phi' ...
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2answers
51 views

$\langle u,\phi\rangle=0$ for all $\phi$ implies $u=0$

Proving the injectivity of a function, I arrived to $\forall \phi\in H^s \int_{\mathbb{R}^d} u(x)\phi(x)dx=0$, where $u\in H^{-s}$. I know that if $\phi\in S$ and $u\in S'$, by the definition of the ...
0
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0answers
26 views

How can I show that $W^{-1,1} \subset W^{-k,p}$?

I need to show that $W^{-1,1} \subset H^{-k}$, for a suitible $k$. Is there anything like a Sobolev embedding theorem for negative Sobolev spaces? I don't think an argument with dual spaces is going ...
3
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1answer
21 views

Sketch of proof: D (Cinf compact supp) dense in H^s

Let $s$ be any real number. $D$ is dense in $H^s$. $D$ is the space of $C^\infty$ functions with compact support. $u\in H^s$ means, by definition of this space of functions, $\hat{u}\in ...
0
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1answer
37 views

Density in sobolev spaces?

Is the Sobolev space $H_0^1(I)$ dense in $L^2(I)$, where $I\subset\mathbb R$? If so, how do I prove it?
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0answers
11 views

Are solutions of nondegenerate PDE positive if the data are the right sign?

Let $\Omega$ be a bounded domain and take Lipschitz function $f:\mathbb{R} \to \mathbb{R}$ be smooth and $0 < c_1 \leq f' \leq c_2$. With the equation $$u' - \Delta f(u) = g$$ $$u(0) = u_0$$ with ...
0
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1answer
29 views

Kind of Cauchy-Schwarz inequality

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Define the Hilbert space $$ H(div;\Omega):=\{u\in (L^2(\Omega))^3:\nabla\cdot u\in L^2(\Omega)\} $$ equipped with the graph norm $$ ...
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0answers
23 views

Zero extension of a $W^{2,\infty}$ Sobolev function outside its domain

Let $O$ be a non empty open subset of a bounded open set $\Omega\subset \mathbb{R}^n$ and let $f\in L^2(\Omega)\cap W^{2,\infty}(O)$. Let $u: \Omega \to \mathbb{R}$ be a function such that the ...
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1answer
31 views

About weak convergence in Sobolev space $W^{1,p}(U)$

The usual definition for a sequence $u_k $ converge weakly to $u$ in $W^{1,p} (U)$ is that $ u_k \rightharpoonup u$ in $L^p (U)$ and ${u_k}_ {x_i } \rightharpoonup u_{x_i }$ in $L^p (U)$ for all $i$. ...
1
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1answer
26 views

Additional sobolev regularity from laplace

Given a bounded Lipschitz domain $U\subset \mathbb{R}^3$ and a function $u\in W^{2,2}(U)$ with $\Delta u\in L^p(U)$ for some $p>2$, does $u\in W^{1,p}(U)$ hold?
5
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1answer
58 views

Is there a Rellich-Kondrachov theorem for manifolds with boundary?

As special case, consider the cylinder $C=[0,T]\times S^n$. Is there a compact embedding $H^1(C)\subset\subset L^2(C)$? The Wikipedia entry to the Rellich-Kondrachov theorem claims that such an ...
0
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1answer
135 views

domain of heat semigroup,

In Example 7.20 p.99 of enter link description here It was stated $D(A)=W^{2,p}$. Could anyone give a proof of above claim? Here, \begin{equation*} D(A) :=\left\{v\in L^p(\mathbb{R}) : \lim_{t\to ...
5
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1answer
102 views

Does an extension operator in Sobolev spaces commute with derivative operators?

Assume that $\Omega\subseteq \mathbb R^d$ is open and has a Lipschitz boundary. Let $\tau\geq0$. Then we know that there exists a linear operator $E:H^\tau(\Omega)\to H^\tau(\mathbb R^d)$ such that ...
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1answer
33 views

A sobolev function $u$ that $u\in L^1$ but $\nabla u\in L^2$. Will it be $H^1$?

Take $\Omega\subset \mathbb R^N$, open bounded, smooth boundary. Take $u_n\subset L^1$ a sequence of functions so that $u_n\to u$ strongly in $L^1$ and $$ \sup_n\int_\Omega |\nabla ...
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0answers
16 views

The existence of solution of wave equation with initial/boundary data in $H^{-1}$ and $L^2$

In a paper by Zuazua, "Propagation, Observation, Control, and Numerical Approximation of Waves", which you can find here http://www.sissa.it/fa/am/DCS2003/reading_mat/zuazuajpg.pdf he considers the ...
0
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1answer
37 views

mollifier for functions in $L^1_{\hbox{loc}}(\Omega)$

The following is from the appendix in Leonin's First Course in Sobolev Spaces: Here is my question: Could anybody explain what is the meaning of "$u_\varepsilon$ is well-defined"? What is ...
2
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0answers
58 views

Embedding of $W^{d, 1}(\overline{\Omega)}$ into $C(\overline{\Omega})$

I've been trying to prove the following assertion: Assume that $\Omega\in C^{0,1}(\mathbb{R}^d)$. Prove that $W^{d,1}(\Omega)\hookrightarrow C(\overline{\Omega}).$ My approach: I have proven that ...
0
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1answer
21 views

Convergence of line restrictions of Sobolev functions

The following result should be true, at least I think I saw it somewhere before but I can't find it now. Please help me to find a reference, or point out if you don't think it is true. Given ...
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0answers
22 views

A limit about measure set

Assume $\Psi\in W^{1,\frac{3}{2}}_{loc}(\mathbb{R^2})$, satisfies: $$\lim\limits_{r\to 0}\frac{1}{r}\int_{B_r(x)}|\nabla\Psi(y)|^{3/2}dy=0$$ Then for each $(x_0,y_0)$, and for every $\epsilon>0$ ...
3
votes
2answers
40 views

Weak formulation with Dirichlet boundary conditions

Let's consider the one-dimensional ODE: $$ u_{,xx}(x)+1=0,\quad \forall x\in ]0,\ell[$$ together with the two Dirichlet boundary conditions $u(0)=0$ and $u(\ell)=0$. The corresponding weak ...
0
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1answer
19 views

Stream Function in Sobolev Space

Assume $\Omega$ is a multi-connected bounded domain in $\mathbb{R}^2$, and function $v\in W^{1,2}_{0}(\Omega)$ satisfies: $$\text{div} \; v=0$$ Then there exists a stream function $\psi\in ...
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0answers
17 views

Diagonalizing operators with the aid of wave packet transformation?

In a class on wave packet transformation, our teacher gave the definition of wave packet transformation $W^h\colon L^2(\mathbb R^n)\to L^2(\mathbb R^{2n})$ for $h>0$ as: $$W^hf(p,q)=\left(\frac ...
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0answers
24 views

Surface gradient definition

Let $\Omega$ be a bounded domain with $C^2$ connected boundary $\partial\Omega$. For a function $p\in H^1(\partial\Omega)$, we define the surface gradient $\nabla_{\partial\Omega}$ as $$ ...
1
vote
1answer
42 views

Use of regularity of the PDE solution to argue the smoothness of a function

I asked the question below before. $\Delta u$ is bounded. Can we say $u\in C^1$? I thought I understood the discussion using the PDE theory at the time but now I am lost. I am going to a similar but ...
1
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1answer
20 views

The upper bound of $L^2$ norm of the minimizer in an minimizing problem.

I am considering the following minimizing problem: $$ u_m:= \operatorname{argmin}_{u\in BV(\Omega)}\{ \frac{1}{2} \|u-u_0\|_{L^2}^2 + t |u|_{TV}\} $$ where $u_0\in BV(\Omega)\cap L^\infty(\Omega)$ and ...
2
votes
0answers
52 views

Is the image of gradient map from Sobolev space to Lebesgue space weakly closed?

Suppose f is a map defined between $W_0^{1,p}(\Omega)$ and $L^{p'}(\Omega)$ as follows - $u \mapsto |\nabla u|^{p-1}$. Is the range of this map weakly closed in $L^{p'}$?.