All Questions
11 questions with no upvoted or accepted answers
3
votes
0
answers
285
views
Error estimate on convolution of mollifiers
Given $u\in W^{1}_{p}(\omega)$ with $1\leq p\leq \infty$, and the mollifier $\rho\in C_0^{\infty}(R^d)$ with support $B_1$ is a unit ball centered at the origin, $\rho\geq 0$ and $\int_{B_1} \rho = 1$....
- 133
2
votes
0
answers
55
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 ...
- 420
2
votes
0
answers
103
views
A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 1,759
2
votes
0
answers
190
views
Inequality on the dual space of $H^s$
Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ?
For instance, assume ...
- 93
2
votes
0
answers
94
views
From some priori estimates can we estimate higher Sobolev norm?
Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that
$$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$
where $u|_{\partial\Omega}=\phi$.
Can we ...
- 309
2
votes
0
answers
274
views
Gagliardo-Nirenberg inequality for periodic functions?
I am interested in a Gagliardo-Nirenberg type inequality for functions in the space
$$H^1_T(\mathbb{R}^n)=\lbrace \psi=u+iv\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)\rbrace$$
...
- 383
2
votes
0
answers
2k
views
Reference for a proof of the Gagliardo-Nirenberg Interpolation Inequality?
In the book Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Cherrier and Milani, Theorem 1.5.2, we are given the following version of the GN interpolation inequality:
Let $\Omega\...
- 1,247
1
vote
0
answers
78
views
Inequality for fractional power norm (sectorial operators)
How could we prove following inequality:
$\int\limits_{0}^{l} u^{3}(x) dx \leq \sqrt{l} \cdot|| u||_{\frac{1}{2}}^{3}$
where
$ || u ||_{\frac{1}{2}} = ||A^{\frac{1}{2}}(u)||_{L^{2}} + || u ||_{L^{...
- 31
0
votes
0
answers
211
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
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
0
votes
0
answers
78
views
Sobolev trace inequality with exterior domains
Let $x_1\in \mathbb{R}^n$, $n\geq 3$, $\Omega=\mathbb{R}^n\backslash B_1(x_1)$, define $D_{\Omega}$ by taking the closure of $C_{c}^{\infty}(\overline{\Omega})$ under the norm
\begin{align*}
\|u\|_{D_{...
- 471