19
votes

Accepted

### Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?

This is true, $f$ is actually everywhere differentiable. It is the "Limit under the Sign of Derivative" Theorem; it also holds for sequences of maps between Banach spaces (and you may even ...

18
votes

Accepted

### Research topics in distribution theory

While I do not know much about current development of the general theory of distributions, I can say something about the current research topics in a special class of distributions, the theory of ...

18
votes

Accepted

### Exercise 8.13 - Brezis

The proof is not short, because it is done from first principles, without using any theorems about Sobolev space except its definition.
By the definition of $W^{1,p}$, there exist $v_n \in C_{c}^{1}(\...

14
votes

Accepted

### Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

That is true. Caccioppoli sets are also known as sets of finite perimeter.
Theorem. Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the ...

13
votes

### Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?

It is a result of Pakzad's that $W^{2,2}$ isometric immersions $f$ of bounded regular convex domains with Lipschitz boundary $\Omega\subset\mathbb{R}^2$ must be developable; more precisely, for every ...

13
votes

Accepted

### Famous but unavailable paper of Jan Boman

I went and scanned it in our library. Here's a Dropbox link.
https://www.dropbox.com/s/ks9gdgi0xwl5j65/Boman%20-%20Lp-estimates%20for%20very%20strongly%20elliptic%20systems.pdf?dl=0

13
votes

### Open problems in Sobolev spaces

Let $H^{s,p}(\mathbb{R}, \mathbb{C})$ be the fractional order Sobolev space of scalar valued functions (distributions) over the real line, where $s\in \mathbb R$ and $1<p<\infty$.
It is a ...

Community wiki

13
votes

Accepted

### Is there any nontrivial characterization of weakly differentiable functions?

Definition.
If $U\subset\mathbb{R}$ is open, we say that $u\in {AC}(U)$
if $u$ is absolutely continuous on every compact interval in
$U$. Let $\Omega\subset\mathbb{R}^n$. We say that
$u$ is absolutely ...

13
votes

### Is there any bilinear Poincaré/Sobolev inequality?

Note: My answer was posted before the question was edited to a different question. My counterexample still works for edit 6 of the question.
Let $\Omega=(0,2)$. Let $$\DeclareMathOperator{\dL}{d\!}u(x)...

12
votes

### Arzelà-Ascoli theorem and Hölder spaces

For completeness, let's mention a simpler and more general statement: For $\Omega\subset\mathbb{R}^n$ a bounded open set, $k\in\mathbb{N}$ and $0<\beta<\alpha\le1$ there is a compact embedding
$...

11
votes

Accepted

### Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior?

Not necessarily- let $\Omega = B_1 \cap \{x_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $H^1_0(\Omega) \cap C^{\infty}(\Omega),$
but $u$ is discontinuous at the origin.

11
votes

Accepted

### Sobolev spaces of differential forms and regular atlases

Ok this is already quite a mouth full, so let me try to give answers to some of your questions:
The main issue is that Sobolev mappings are defined via a boundedness concept (you ask for $L^p$-...

11
votes

Accepted

### Chain rule in Sobolev space

The main difficulty in the proof of the rule is to prove that $\nabla u=0$ a.e. on the set $u^{-1}(\Sigma)$, where $\Sigma$ is the set where $F$ is not differentiable; and where $\nabla u=0$ one ...

10
votes

### Books about capacity theory

I think the best treatment of basic facts about capacity from the perspective of Sobolev spaces is in Chapter 4 of
L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised ...

10
votes

Accepted

### Can we approximate any open set by sub-domains with smooth boundary?

By a well-known theorem of Whitney, any closed subset of $R^n$ coincides with the zero set of a $C^\infty$ function:
Whitney, Hassler: Analytic extensions of differentiable functions defined in ...

10
votes

### Comparison of two versions of fractional Sobolev spaces: do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?

This is standard and well explained in several treatises: the space $W^{s,p}$ belongs to the scale of Besov spaces, while $H^{s,p}$ is in the scale of Triebel-Lizorkin spaces. The two scales satisfy ...

10
votes

### Research topics in distribution theory

I think this is a good question. I am no longer working in this field. When I was working in this field, there is no general program and I do not know what are the important open problems. Instead of ...

Community wiki

10
votes

### Sobolev spaces of differential forms and regular atlases

There is a coordinate free way of defining Sobolev spaces of sections of a vector bundle $E$ over a manifold $M$. You need to make a few choices: a metric $g$ on $M$, a metric $h$ on $E$ and a ...

10
votes

Accepted

### Possible way to define $H_0^1(\Omega)$ Sobolev spaces

The first two are equivalent, as the $H^1(\Omega)$ norm and $H^1(\mathbb{R}^d)$ norm coincide for $C^\infty_c(\Omega)$ functions.
The third is in general different:
If you let $d = 1$ and $\Omega = \...

9
votes

Accepted

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am somewhat confused that, despite saying many true and useful things, no one has said directly that $H^{-1}$ on the circle can be characterized as the set of distributions $\theta$ such that $\sum_{...

9
votes

Accepted

### Density of polynomials in $C^k(\overline\Omega)$

No, the polynomials will not be dense in general.
The following example is essentially one-dimensional. Let $C\subset[0,1]$ be the usual ternary Cantor set and $g\colon[0,1]\to[0,1]$ the Cantor ...

9
votes

### Does anyone know what is the right reference for the following simple lemma from harmonic analysis?

This inequality is also a corollary of the main result of
Fefferman, Charles; Stein, Elias M., Some maximal inequalities, Am. J. Math. 93, 107-115 (1971). ZBL0222.26019.
which asserts that
$$ \| \...

9
votes

Accepted

### The Hölder inequality for fractional order Sobolev seminorm?

Your question can be rephrased by asking whether one has a HÃ¶lder estimate
$$
|u|_{W^{s, p}} \le C |u|_{W^{s, q}},
$$
when $p < q$ or whether $W^{s, q} \subset W^{s, p}$.
There is no such ...

9
votes

Accepted

### Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?

The density result is true for any family of vector fields with Lipschitz coefficients.
Theorem. Let $X_1,\ldots,X_k$ be a system of vector fields with Lipschitz coefficient on a compact ...

9
votes

### A characterization of constant functions

Not quite an answer, but too long for a comment.
Let me make my life easier a bit and take $\Omega=\mathbb{R}^N$ while increasing the exponent slightly. Namely, I will assume that
$$
I:=\ \int_{\...

9
votes

Accepted

### Arzelà-Ascoli theorem and Hölder spaces

At first, if partial derivatives of order at most $k$ of $f_{n_i}$ converge to those of $f$, than automatically $f\in C^{k,\alpha}(B)$, since $$|(D^k f)(x)-(D^k f)(y)|\leqslant \limsup_i |(D^k f_{n_i})...

9
votes

Accepted

### Initial conditions in the Klein-Gordon equation

One must remark that derivatives in Sobolev spaces are usually taken in the sense of distributions: given $k\in\mathbb{N}_0=\{0,1,2,\ldots\}$, $H^k(\mathbb{R}^n)$ is the space of tempered ...

9
votes

Accepted

### Eigenvalues and eigenfunctions of the Laplace operator on entire plane

The point spectrum coincides with the spectrum minus 0 if $p>2n/(n-1)$ and it is empty in the remaining cases ($n$ is the dimension). This is proved in G. Talenti: "Spectrum of the Laplace ...

9
votes

Accepted

### Best constant for Poincaré inequality on spheres

The best constant is just the multiplicative inverse of the smallest positive eigenvalue of the Laplacian on the sphere. On $\mathbb{S}^N$ this the smallest eigenvalue is $N$, so $C(N)$ in that case ...

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