All Questions
253 questions
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
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
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
14
votes
2
answers
536
views
Reference Request: Elliptic differential operators in the Fréchet setting
Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
- 5,824
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
13
votes
3
answers
2k
views
Sobolev spaces and geometry
This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study PDE'...
- 947
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}(\...
- 29.8k
12
votes
0
answers
476
views
Are Sobolev trace spaces equal from both sides of the boundary?
Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
- 8,086
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
10
votes
1
answer
973
views
$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$
I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
- 261
10
votes
1
answer
2k
views
Chain rule for distributional derivative
Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$).
Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
- 145
9
votes
1
answer
639
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
- 807
9
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
9
votes
1
answer
1k
views
Sobolev space for Mixed Dirichlet - Neumann boundary condition
Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
- 253
8
votes
1
answer
496
views
Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?
Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space:
$$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
- 287
8
votes
1
answer
496
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
- 4,135
8
votes
0
answers
177
views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
8
votes
0
answers
260
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 503
7
votes
2
answers
3k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 21.8k
7
votes
2
answers
508
views
Making the Fourier transform quantitative
I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...
- 225
7
votes
1
answer
1k
views
Eigenvalues and eigenfunctions of the Laplace operator on entire plane
According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
- 597
6
votes
2
answers
353
views
Bounded deformation vs bounded variation
Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...
- 608
6
votes
2
answers
2k
views
Equivalent Norms on Sobolev Spaces
When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ^{\...
- 233
6
votes
2
answers
799
views
A question on density of Lipschitz functions in weighted Sobolev spaces
Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$.
Let now $...
- 1,881
6
votes
1
answer
173
views
Sobolev space is spanned by distributions supported on half-lines?
I asked this question on Mathematics Stack Exchange previously.
This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it.
For any $s \leq 1/2$,
$$H^s(\mathbb{...
- 86
6
votes
1
answer
246
views
The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients
Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by
$$Lu = \partial_i(a^{ij}...
- 91
6
votes
1
answer
331
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
- 3,477
6
votes
0
answers
113
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
5
votes
2
answers
2k
views
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \...
- 371
5
votes
2
answers
937
views
Second order differentiability of subharmonic function almost everywhere?
The following general definition of subharmonic function comes from the classical text book [elliptic partial differential equations of second order] by Gilbarg and Trudinger.
We call a function $u$ ...
- 1,350
5
votes
3
answers
1k
views
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\...
- 307
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,135
5
votes
1
answer
453
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 1,101
5
votes
1
answer
260
views
Approximate Sobolev embedding
It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define ...
- 124
5
votes
2
answers
1k
views
Compactly supported functions and Sobolev spaces on manifolds
It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
- 9,853
5
votes
1
answer
448
views
improved Sobolev embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding.
...
- 1,385
5
votes
1
answer
481
views
A continuous path between two Sobolev functions
Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...
- 679
5
votes
1
answer
359
views
Alternative representations of Sobolev space
Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...
5
votes
0
answers
438
views
Green's formula and traces in weighted Sobolev spaces
Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let
\begin{equation}
\rho(x) = 1-|x| \quad \text{ for } x \in B_1,
\end{equation}
and let $\sigma >0$ be given. As per the comments, notice that $\...
- 309
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
4
votes
2
answers
781
views
Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
- 185
4
votes
1
answer
458
views
Contractivity of Neumann Laplacian
I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.
In ...
- 1,759
4
votes
3
answers
1k
views
A space of distributions vanishing on the boundary
The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...
- 5,407
4
votes
1
answer
698
views
Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions
Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type
$$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$
to hold for all $u\in X\subset H^1(\Omega)$ and $C$...
- 287
4
votes
3
answers
308
views
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
- 1,759
4
votes
1
answer
398
views
Proving the inequality $|\nabla |\nabla^r \psi|| \le |\nabla^{r+1} \psi|$
Following Aubin's book "Some nonlinear problems in Riemannian geometry", we use the notation
$$
|\nabla^r \psi|^2 = \nabla_{\alpha_1}\cdots \nabla_{\alpha_r}\psi \nabla^{\alpha_1}\cdots \nabla^{\...
- 1,245
4
votes
1
answer
147
views
Embeddings of the maximal domain for the Laplacian
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...
- 41
4
votes
1
answer
1k
views
Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$
I want to show:
Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align}
H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)
\end{align}
is compact.
I was able to show ...
- 437
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