All Questions
Tagged with real-analysis sobolev-spaces
25 questions
5
votes
2
answers
242
views
Can we stay invertible while approximating linear maps in Sobolev spaces?
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.
Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
- 6,741
21
votes
1
answer
3k
views
Density of polynomials in $C^k(\overline\Omega)$
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
- 4,034
16
votes
4
answers
2k
views
Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with
$f_n \to f$ uniformly for some (necessarily) continuous $f$.
$f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
Is it true ...
- 6,155
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
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 1,881
6
votes
1
answer
181
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence of ...
- 6,741
4
votes
1
answer
785
views
What is the dual space of $L^p$(conservative vector fields on a bounded set)?
First, some background: I wanted to prove that, if $f$ is a measurable function such that $\nabla f\in L^p_\text{loc}(\mathbb R^n)$, then $f\in L^p_\text{loc}(\mathbb R^n)$, $p\in(1,\infty)$. This is ...
- 391
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
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
33
votes
1
answer
2k
views
For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
- 43.2k
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
6
votes
1
answer
2k
views
Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold
How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
- 839
5
votes
0
answers
140
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
- 1,101
4
votes
1
answer
92
views
Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary).
Does there exist a sequence of ...
- 6,741
4
votes
1
answer
308
views
Adjoint of the multiplication operator on a Sobolev space
Let $f\colon\mathbb{R}^n\rightarrow\mathbb{C}$ be a bounded function with a bounded first derivative. Then the multiplication operator $H^1(\mathbb{R}^n)\ni x\mapsto A_f x:=fx\in H^1(\mathbb{R}^n)$ is ...
- 128k
3
votes
1
answer
496
views
"Square root" of multiplication operator on Sobolev space
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a non-negative, smooth, uniformly bounded function with uniformly bounded first derivative. Then $f$ defines a bounded operator on $L^2(\mathbb{R}^n)$ as ...
- 1,903
2
votes
1
answer
101
views
Convergence of energy of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by
\begin{equation}
f_r(x):=
\begin{cases}
...
- 169
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
1
vote
2
answers
183
views
Convergence of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$.
Question 1: Can we ...
- 169
1
vote
1
answer
201
views
Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
1
vote
1
answer
110
views
Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$
Let $\Omega
\subset
\mathbb{R}^{N}$
be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Caratheodory function such that $g(x,t)=0$
for $t\leq0$
. Suppose that ...
- 135
0
votes
1
answer
110
views
Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 131
0
votes
1
answer
491
views
Is this set of function belongs to $L^\infty$?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
- 679
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
-1
votes
1
answer
208
views
Does this function belong to $L^2(\mathbb{D})$?
Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question.
The unit ...
- 356