All Questions
10 questions
11
votes
1
answer
1k
views
The Hölder inequality for fractional order Sobolev seminorm?
This question is post on MSE a week ago. I move it here to draw more attention.
Let $u\in C^\infty(\bar I)$ be given where $I=(0,1)$. Define
$$
t(\alpha):=\left(\int_I\int_I \frac{|u(x)-u(y)|^\alpha}{...
- 679
4
votes
1
answer
388
views
Dependence of the constant in Korn's inequality on the domain
Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and
$$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i
j} ( v) \varepsilon_{i j} (...
- 2,222
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
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
1
answer
140
views
interpret of Picone inequality for non-regular functions
Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
- 371
2
votes
0
answers
56
views
Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
- 1,101
2
votes
0
answers
250
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 197
1
vote
1
answer
148
views
Understanding a family of Sobolev-type inequalities
I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:
Denote the following inequality as $S_{r,s}^{\theta}$: $\...
- 11
0
votes
1
answer
160
views
Global Poincaré type estimate
For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...
- 4,135
0
votes
0
answers
149
views
Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477