All Questions
Tagged with fa.functional-analysis sobolev-spaces
652 questions
4
votes
4
answers
4k
views
Variation on the Sobolev space $H^1_0$
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\},
$$
and let $C^1_c(\Omega)$ be the space of ...
- 3,322
2
votes
1
answer
253
views
Isocapacity inequalities in the theory of Sobolev Spaces
Section 9 of the lectures notes of Maz'ya (Мазья) on isocapacity, lectures notes that can be found at:
http://www.math.liu.se/~vlmaz/pdf/mazya.pdf,
discusses inequality between the $L^p$ norm in a ...
- 207
22
votes
4
answers
3k
views
When to use more exciting function spaces than ordinary Sobolev spaces?
In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces $L^2(0,T;H^...
- 405
2
votes
1
answer
988
views
Compact embedding of weighted sobolev spaces in continous functions spaces
Let the weighted sobolev space $M^p_{s,\delta}$ be the completion of $C_0^\infty(\mathbb{R}^n)$ in the norm \begin{equation}
\sum_{\left\vert\alpha\right\vert\leq s}\left\Vert(1+\left\vert x\right\...
- 327
3
votes
0
answers
201
views
Degree of a function in $H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$
We can define the degree of a function $f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$ as
$$\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial \...
- 31
2
votes
1
answer
208
views
Does a particular iteration produce a weak solution to a non linear pde?
Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...
- 3,245
5
votes
0
answers
428
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
5
votes
0
answers
543
views
Moduli of smoothness, Besov spaces, and Sobolev spaces
For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}
\end{equation}
where $\Omega_{rh}=\{...
- 3,322
14
votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
- 24.7k
2
votes
0
answers
252
views
compact embedding in duals of weighted Sobolev spaces
On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding
$$
L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset (W^{k,q}(\mathbb{R}^d,\...
- 5,380
0
votes
1
answer
326
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 $L^2$?
...
- 63
1
vote
0
answers
305
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 u\|...
- 51
6
votes
1
answer
498
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 ...
- 1,714
1
vote
1
answer
133
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 ...
- 13
1
vote
2
answers
444
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 ...
- 11
1
vote
0
answers
618
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 $0<r<\frac{2(n-s)...
- 11
4
votes
2
answers
228
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 - y\partial_x)...
0
votes
0
answers
160
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 $...
- 1
4
votes
1
answer
4k
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 ...
- 73
2
votes
1
answer
1k
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
$$H^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\...
- 23
-1
votes
1
answer
187
views
Limit of a function in a weighted Sobolev space
I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense
$$\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} f(x)=...
- 25
2
votes
1
answer
412
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 $$f(x)=\...
- 21
0
votes
0
answers
214
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 ...
- 101
10
votes
2
answers
6k
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 $...
- 882
3
votes
1
answer
785
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) ...
- 3,388
3
votes
1
answer
2k
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)$, ...
- 31
11
votes
3
answers
1k
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 matter....
- 882
9
votes
1
answer
758
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) ...
- 2,998
3
votes
1
answer
6k
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 $...
- 895
9
votes
4
answers
2k
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 ...
- 2,222
1
vote
2
answers
488
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} } \right\|_{L^q(\mathbb{R}^n)...
- 13
4
votes
2
answers
904
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 ...
- 617
7
votes
1
answer
659
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 ...
- 4,000
2
votes
1
answer
382
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 ...
0
votes
1
answer
2k
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 I$...
3
votes
1
answer
393
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 |f''|^2]...
- 143
12
votes
3
answers
2k
views
Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.7k
0
votes
1
answer
717
views
Interpolation of derivatives
If $U$ is an open interval of $\mathbb{R}$ and $f : U \to \mathbb{R}$ is an $L^2(U)$ function with second derivative $f'' \in L^2(U)$ (in the weak sense), is $f \in W^{1,1}(U)$?
EDIT: Removed false ...
- 3
3
votes
1
answer
571
views
Subspaces of a Sobolev space
For $a \in \mathbb{R}^N\setminus\{0\}, N \ge 2$, and $\lambda \in \mathbb{R}$ let
$$
X_{\lambda,a}=\{u(\cdot+\lambda a):\, u(x)=u(|x|) \in W^{1,2}(\mathbb{R}^N)\}.
$$
Denote by $X_a$ the closure of ...
- 397
27
votes
2
answers
8k
views
Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp
Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...
- 309
4
votes
1
answer
471
views
Embeddings for spaces of maximal regularity
Let $T\in(0,\infty)$ and $\Omega\subset\mathbb R^n$ be a smooth domain. In terms of maximal regularity it can be very beneficial to know for which $s_i,p,n$ the following holds true
$W^{s_1,p}(0,T;L^...
- 225
22
votes
1
answer
4k
views
Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
- 39k
0
votes
2
answers
444
views
Sobolev space: probably simple ode....
I am trying to solve for $y(x)$ in terms of $f(x)$ in a convenient space (eg. $\dot{H}^2(\mathbb{T})$-zero mean). Here is the ode:
$y(x)+y(x)y'(x)=f(x)$.
I think a contraction mapping argument will ...
- 9
0
votes
2
answers
273
views
Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide?
Let $I$ be a bounded interval and consider a sequence $(u_k)$ in $H^{1,2}(I)$ (usual Sobolev space). Suppose furthermore, that the sequence $(u_k)$ is bounded in $H^{1,2}(I)$. Then, by Rellich, we can ...
- 2,935
5
votes
2
answers
1k
views
Sobolev imbedding on Riemannian manifolds
Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.
Let $W^{1,...
- 397
2
votes
1
answer
431
views
Sobolev imbedding
It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with
$n_1+...
- 397
9
votes
3
answers
2k
views
Trace theorem for $C^{k,1}$ domains
What are the best results on (Sobolev space) trace theorems for $C^{k,1}$ domains?
For $k=0$, e.g., when the domain is Lipschitz, from e.g. the works of Martin Costabel and Zhonghai Ding, it is known ...
- 3,322
4
votes
1
answer
1k
views
Sobolev-Slobodeckij spaces for p=infinity
For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,
$
\...
- 3,217
0
votes
0
answers
362
views
Gradient of the energy functional in $H^{1,2}$-norm
I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
- 2,935
4
votes
2
answers
4k
views
Embedding of $BV$ and $L^p$ spaces
An elementary question about Sobolev spaces:
Is there some explicit theorem about embedding relation between spaces $BV(\Omega)$ and $L^p(\Omega)$?
Formulated otherwise: is $BV$ a subset of $L^2$ (i....
- 53