The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
166 views

Is BV2 space closed in L2 space?

We define the BV2 space by $S = \lbrace f\in L^2:\textrm{TV}(f)<\infty\rbrace$, where $TV(f)=\sup_{g\in C_c^1,\|g\|_\infty\leq 1}\int f\cdot \textrm{div}g$. My question is: is $S$ closed in $L^2$? ...
0
votes
0answers
143 views

Adjoint operator in sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta ...
5
votes
0answers
168 views

A finely open set, not open up to polar set?

I already asked this on M.SE, but get no answers. Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
-1
votes
1answer
101 views

A suitable Sobolev-type space

Let $\Omega$ be a domain in $\mathbb{R}^3$, does there exist a vector-valued Sobolev-type space, or maybe space in other sense, $V$, satisfying the following: 1) $S:=\lbrace v\in ...
5
votes
1answer
392 views

Compactness in Sobolev spaces

I was wondering whether the set $\lbrace f\in H_0^1(\Omega)|\|f\|_{L^\infty(\Omega)}\leq 1\rbrace$ is compact in $H_0^1(\Omega)$ or not. Here $\Omega$ is a convex domain in $\mathbb{R}^3$ with ...
5
votes
1answer
291 views

Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions

This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f ...
1
vote
1answer
103 views

Nonintegrable inverse powers as distributions

I am working through Lieb/Loss's "Analysis", and have been stuck on one of the problems for a while; Suppose we are on $\mathbb{R}^n$ and define $f(x) = |x|^{-n}$. This is not a locally integrable ...
4
votes
1answer
237 views

Sharpness of the Sobolev embedding theorem

We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, ...
1
vote
0answers
126 views

Trace Inequality question

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
1
vote
2answers
284 views

A question on optimal Sobolev inequality.

Let us consider the Sobolev inequality $||u||_{L^p} \le C||u||_{H^1}$ for $2 <p< 2^*$, where the constant $C$ depends on $p$ and the domain. My question is, how can one see that the optimal ...
1
vote
0answers
311 views

A question on a variant of Hardy's inequality.

I would like to know a proof of a variant of Hardy's inequality below. Could anyone introduce me a reference or give me a proof? Thank you very much for your assistance. Set ...
4
votes
1answer
214 views

projection of sobolev spaces onto cones

Consider the Sobolev space $W^{k,p}(\Omega)$ for $k\in \mathbb N$, $p\in [1,\infty]$ and some open domain $\Omega\subset \mathbb R^n$ $^*$. Then it is known that $W^{k,p}(\Omega)$ is an ordered Banach ...
4
votes
2answers
221 views

Finding an optimal $p$ such that $u \in L^p$

We have an $L^2$ function $u$ defined on $\mathbb{R^2}$ with compact support such that $u \in H^{2/3}$ (H stands for Sobolev spaces, as always), $\partial_y u \in L^2$, and $(x\partial_y - ...
0
votes
0answers
122 views

Is this function in the weighted Sobolev space $H^{2,-s}$?

I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order ...
2
votes
1answer
478 views

Doubt on norm of the Sobolev space $H^2(\mathbb{R}^3)$

If I consider the Sobolev space $H^2(\mathbb{R}^3)$ I have the norm $$\Vert u\Vert_{H^2(\mathbb{R}^3)}=\bigg(\sum_{|\alpha|\leq 2}\Vert D^\alpha u\Vert^2_{L^2(\mathbb{R}^3)}\bigg)^\frac{1}{2}.$$ Is ...
0
votes
1answer
526 views

Embedding of weighted Sobolev spaces

I define the following wheighted Sobolev spaces $$L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$$ and $$H^{2,s}(\mathbb{R}^3)=\bigg\lbrace ...
-1
votes
1answer
142 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} ...
-1
votes
1answer
88 views

Every function in W^{1,1}(0,1) is continuous on (0,1)

I am trying to prove that if $u:(0,1)\to\mathbb{R}$ lies in $W^{1,1}(0,1)$, then $u\in C(0,1)$. Is there any help anybody can offer? Thanks.
22
votes
1answer
1k views

What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...
2
votes
1answer
214 views

Convergence in norm of Sobolev spaces

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\lbrace{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\rbrace}$ and I have to show that the function ...
0
votes
0answers
147 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
0
votes
2answers
632 views

Interior regularity for elliptic equations

The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for ...
8
votes
2answers
1k views

Characterizing the Dual of $W_0^{s,p}$

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and ...
2
votes
3answers
818 views

How to define Laplacian on $L_2$

This might be a dumb question, but I thought the Laplacian (classical) is defined for $C^2$ functions. How do we extend that to be a self-adjoint operator on all of $L_2$? Is it the so called ...
3
votes
1answer
295 views

on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
2
votes
1answer
664 views

Norm of differential operator between Sobolev spaces

It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms) $W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...
9
votes
3answers
408 views

Boundedness of the derivative of the trace of an H^1 function

As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical ...
7
votes
1answer
241 views

Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
3
votes
1answer
2k views

Classical Derivative, Weak Derivative and Integration by Parts

Hello, While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated. QUESTION Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function ...
1
vote
1answer
65 views

Continuity of an extension map

Suppose $\delta\in (0,1)$ and $r<1+\delta.$ Suppose moreover we are given a sequence of functions $u_m\in H^{1/2,2}(\partial B_r(0))$, where $B_r(0)$ denotes the euclidean $n-$dimensional ball. ...
3
votes
2answers
184 views

Regularity properties of H(-1/2)

Arising as the traces of $H(div; \Omega)$, I am wondering if the space $H^{-1/2}(\partial \Omega)$ has any regularity properties? (Containment in BV would be wonderful, although I doubt it holds.) It ...
5
votes
1answer
420 views

Books about Capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
3
votes
0answers
198 views

$\tilde{u}=0,\ a.e.\ x\in\Gamma$?

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in ...
2
votes
1answer
344 views

Weak divergence implies weak differentiability of components?

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$. We say that $\sigma$ has weak divergence if there exists ...
1
vote
2answers
406 views

Sobolev-type inequality.

Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $ \left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } ...
6
votes
4answers
646 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by \begin{equation} ...
3
votes
2answers
379 views

Nash inequality on a compact domain?

I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...
6
votes
1answer
310 views

Compactness of Sobolev embedding for domains of finite measure

Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero ...
2
votes
1answer
397 views

Definition of Sobolev spaces as a space of sections of certain type

I want to define Sobolev spaces for sections on a vector bundle, basically I want that a section will belong to the Sobolev space $W^{k,p}$ if its coordinates in any aceptable patch belong to the ...
3
votes
2answers
245 views

Boundary Value Problem in the space of Distributions

I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the ...
7
votes
1answer
597 views

Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
2
votes
1answer
286 views

Function extension in a Sobolev space

Let $\Omega$ be a domain of $R^n$ and let $H^2(\Omega)$ be the usual Sobolev space. Let $\emptyset\ne \omega_1\subset\omega_2$ be open subsets of $\Omega$, and let $\theta \in H^2(\omega_1)$. I ...
1
vote
0answers
342 views

Relation between interpolation spaces and besov spaces

Consider the following two norms: The interpolation norm: 1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...
6
votes
1answer
151 views

$W^{1,1}$ simplicial approximation

Let $f$ be a continuous real-valued function defined on an $n$ dimesional simplex $\Sigma\subset \mathbb{R}^n $. The classical simplicial approximation scheme provides a sequence $f_k$ of piecewise ...
-1
votes
1answer
745 views

weak convergence in Sobolev spaces and pointwise convergence?

I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that $\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup ...
4
votes
2answers
830 views

Example for the Sobolev embedding theorem when p=n.

Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then $u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 ...
1
vote
1answer
486 views

A property on the Green-St Venant strain tensor

Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient. Show that $u\in H^1(\Omega), E(u)\in ...
3
votes
2answers
524 views

Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $

The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ? For detail : Fix $p \geq 1. ...
2
votes
1answer
118 views

Second derivative seminorm of function composition

Consider the seminorm $\| f \|^2 = \int_{-\infty}^{\infty} dx f''(x)^2$ for $f:\mathbb{R}\rightarrow \mathbb{R}$ in the Sobalev space $W^{k,2}(\mathbb{R})$. Can we put some upper bound on the ...
3
votes
1answer
319 views

A Sobolev-type inequality with weights

In the study of a particular PDE I found myself wanting to prove the following inequality: $( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r ...