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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 $\Omega$, ...
user111's user avatar
  • 4,034
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}(\...
Piotr Hajlasz's user avatar
4 votes
1 answer
786 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 ...
Lentes's user avatar
  • 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^...
Iosif Pinelis's user avatar
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^...
André Henriques's user avatar
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 $K$. ...
Gil Kalai's user avatar
  • 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?
Riku's user avatar
  • 839
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 ...
Iosif Pinelis's user avatar
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 ...
geometricK's user avatar
  • 1,903
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^\...
dohmatob's user avatar
  • 6,853
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 ...
user109584's user avatar
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-...
JumpJump's user avatar
  • 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 ...
Isaac's user avatar
  • 3,477
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 ...
Hiro's user avatar
  • 131