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19 votes
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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 ...
Pietro Majer's user avatar
18 votes
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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 ...
Piotr Hajlasz's user avatar
18 votes
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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}(\...
Yuval Peres's user avatar
  • 14.1k
14 votes
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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 ...
Piotr Hajlasz's user avatar
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 ...
j.c.'s user avatar
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13 votes
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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
Dan Petersen's user avatar
  • 39.5k
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 ...
13 votes
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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 ...
Piotr Hajlasz's user avatar
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)...
user479223's user avatar
  • 1,611
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 $...
Pietro Majer's user avatar
11 votes
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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.
Connor Mooney's user avatar
11 votes
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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 $L^p$-...
Alexander Schmeding's user avatar
11 votes
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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 ...
Piero D'Ancona's user avatar
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 ...
Piotr Hajlasz's user avatar
10 votes
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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 ...
Mohammad Ghomi's user avatar
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 ...
Piero D'Ancona's user avatar
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 ...
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 ...
Liviu Nicolaescu's user avatar
10 votes
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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 = \...
Willie Wong's user avatar
  • 38.7k
9 votes
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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_{...
paul garrett's user avatar
  • 22.7k
9 votes
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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 ...
Manfred Sauter's user avatar
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 $$ \| \...
Terry Tao's user avatar
  • 112k
9 votes
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The Hölder inequality for fractional order Sobolev seminorm?

Your question can be rephrased by asking whether one has a Hölder 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 ...
Jean Van Schaftingen's user avatar
9 votes
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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 ...
Piotr Hajlasz's user avatar
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_{\...
Abdelmalek Abdesselam's user avatar
9 votes
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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})...
Fedor Petrov's user avatar
9 votes
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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 ...
Pedro Lauridsen Ribeiro's user avatar
9 votes
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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 ...
Giorgio Metafune's user avatar
9 votes
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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 ...
Willie Wong's user avatar
  • 38.7k

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