All Questions
77 questions
1
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Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\...
- 1,101
2
votes
0
answers
54
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 410
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
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
2
votes
2
answers
197
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$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
3
votes
1
answer
296
views
Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...
- 677
2
votes
0
answers
124
views
Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
- 21
7
votes
1
answer
652
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Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
- 820
2
votes
2
answers
451
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The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
4
votes
2
answers
391
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 5,380
3
votes
1
answer
466
views
Equivalence between two fractional Sobolev spaces
For $s \in (0,1)$, we consider the spectral fractional Laplacian
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
...
- 161
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
2
votes
1
answer
351
views
References on duality of fractional order Sobolev spaces
I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
- 101
2
votes
0
answers
150
views
Reference for weighted Sobolev spaces
I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
- 235
1
vote
1
answer
404
views
Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? [closed]
Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
- 6,853
1
vote
0
answers
110
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Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
6
votes
1
answer
382
views
Sobolev embedding theorems on manifolds
I had asked the following question on math.stackexchange but did not get any response:
I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
- 131
1
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0
answers
79
views
Reference for smoothness of Nemytskii operator on fractional Sobolev spaces
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator
$$
\big(N_\varphi x\big)(t)=\varphi\big(x(t)\big)
$$
for $x\in H^s(T^d)$, the ...
- 93
2
votes
0
answers
559
views
Multiplication in Sobolev space with negative exponent
My initial problem is the following: I would like to estimate $\lVert f^2\rVert_{H^{-2}}$ in the sense of Sobolev embeddings, where $f:\Omega \rightarrow \mathbb{R}$ is a function defined on a bounded ...
- 914
1
vote
0
answers
56
views
Smooth approximation in Sobolev spaces for surfaces with boundary
Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
- 163
1
vote
1
answer
116
views
uniform convergence of $H^r$ projectors on compact sets?
Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
- 5,380
2
votes
0
answers
125
views
Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system
...
- 1,245
0
votes
1
answer
96
views
Interpolated Sobolev norm inequality
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
- 13
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0
answers
181
views
Variational problems living in two different Sobolev spaces
Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type:
$$\inf_{u,v}\int_{\Omega} ...
- 4,459
5
votes
1
answer
453
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 1,101
2
votes
0
answers
72
views
Product of Besov and Lorentz functions
Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
3
votes
1
answer
1k
views
Friedrichs mollifiers and Sobolev spaces
$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
- 1,141
4
votes
0
answers
176
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If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$
Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
- 51
4
votes
1
answer
256
views
Regularity of Nemitskii maps on Sobolev spaces
Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).
Let $X=W^{1,p}(\Omega)$ with $p>1$ be the ...
- 41
1
vote
1
answer
277
views
Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$
Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...
- 119
6
votes
1
answer
320
views
Stability of fractional Sobolev spaces under diffeomorphisms
Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
- 83
5
votes
2
answers
1k
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Rates of convergence of mollifiers with Sobolev norms on manifold
Let $M$ be a smooth compact Riemannian manifold of dimension $n$, and let $H^s_p(M)$ for $s\in \mathbb{R}$ be the fractional Sobolev space of order $s$ on the manifold (defined for instance through ...
- 83
3
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1
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695
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Continuous embedding between parabolic Sobolev spaces
I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?
Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
- 503
4
votes
0
answers
102
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$W^{k,p}$ and Holder regularity for linear elliptic systems with Neumann boundary data
I'm looking for a text or paper that discusses regularity in the Sobolev and Holder sense for general linear elliptic systems of PDEs on bounded domains with Neumann boundary data.
The book by ...
1
vote
1
answer
219
views
Harmonic functions vanishing on the boundary and distance function asymptotics
Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...
user139845
2
votes
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
4
votes
0
answers
111
views
A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$
Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with
$$
\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u
$$
but couldn't find a good ...
- 1,245
1
vote
0
answers
198
views
Sobolev embedding in complete manifold
Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose that $(M,g)$ admits a bounded geometry.
Q Can we show that for $k-\frac{m}{p}\geq l-\frac{m}{q}$, we ...
- 1,915
2
votes
0
answers
255
views
Sobolev Multiplication on non-compact manifold
We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
- 1,915
2
votes
0
answers
331
views
Sobolev embeddings for vector-valued functions
I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space.
In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
3
votes
1
answer
1k
views
Gagliardo-Nirenberg inequality for bounded domain
For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that
$$
\|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}.
$$
This is also true if $u\in W^{1,2}_0(\Omega)$ for a ...
- 1,245
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
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
5
votes
0
answers
445
views
Why are functions with vanishing normal derivative dense in smooth functions?
Question
Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm?
Here I define $...
- 881
3
votes
1
answer
580
views
What's the predual of Holder continuous function spaces?
Given Holder space $C^{\alpha}(\mathbb{R}^n)$, does there exist a Banach space $X$ such that the dual of $X$ is $C^{\alpha}(\mathbb{R}^n)$?
What I can imagine is that such $X$ must contain the ...
- 1,350
4
votes
1
answer
2k
views
Homogeneous fractional Sobolev spaces
Given $s\in (0,1)$ and a measurable function $u:\mathbb{R^n}\to\mathbb{C}$, let us define $$\|u\|_{\dot H^s(\mathbb{R}^n)}^2:=\iint\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$
and let $\dot H^s(\...
- 3,146
2
votes
0
answers
207
views
Smoothing properties of analytic semigroups
Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates
$\|T_p(t)f\|_{...
- 21
1
vote
1
answer
1k
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Introductory text to Sobolev spaces and PDE's [closed]
I'm looking for a good introductory to Sobolev, preferably with an emphasis to their relationship to PDE's analysis.
I have only seen thus far Giovanni Leoni's "First Course in Sobolev Spaces" which ...
- 3,574
5
votes
0
answers
341
views
Real interpolation of weighted Sobolev spaces with different weights
Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
