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**31**

votes

**1**answer

771 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 ...

**22**

votes

**1**answer

1k views

### What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...

**18**

votes

**4**answers

962 views

### When to use more exciting function spaces than ordinary Sobolev spaces?

In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces ...

**14**

votes

**1**answer

2k views

### Image of the trace operator

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset ...

**12**

votes

**2**answers

291 views

### When is the closed unit ball in a smaller Banach space closed in a larger Banach space?

Recently I saw an interesting lemma:
For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in ...

**12**

votes

**2**answers

3k views

### Compact Embeddings of Sobolev Spaces: A Counterexample Showing The Rellich-Kondrachov Theorem Is Sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...

**12**

votes

**0**answers

417 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 $K$.
...

**10**

votes

**1**answer

199 views

### Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...

**9**

votes

**3**answers

439 views

### Boundedness of the derivative of the trace of an H^1 function

As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical ...

**9**

votes

**1**answer

1k views

### History of Sobolev space notations

I would like to know the historical reason why the letter H is used for Sobolev spaces. In particular, why not S? It would be interesting to know the same for the letter W.

**8**

votes

**4**answers

1k views

### Trace theorem for $C^{k,1}$ domains

What are the best results on (Sobolev space) trace theorems for $C^{k,1}$ domains?
For $k=0$, e.g., when the domain is Lipschitz, from e.g. the works of Martin Costabel and Zhonghai Ding, it is known ...

**8**

votes

**2**answers

740 views

### Extension theory with bump function

Let $B_t(0)$ denote the $n$ dimensional ball of radius $t$ centered at the origin. Does there exist a $\phi\in C(\mathbb{R}^n)$ function with the properties:
$
\phi (x) =
\begin{cases}
1&x\in ...

**8**

votes

**2**answers

1k views

### Characterizing the Dual of $W_0^{s,p}$

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and ...

**7**

votes

**1**answer

764 views

### Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...

**7**

votes

**3**answers

973 views

### Reference request: Simple facts about vector-valued Sobolev space

Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = ...

**7**

votes

**3**answers

412 views

### Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...

**7**

votes

**1**answer

253 views

### Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...

**7**

votes

**2**answers

531 views

### Boundary regularity for the Dirichlet problem

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator.
We wish to solve the Dirichlet problem ...

**7**

votes

**1**answer

367 views

### Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...

**7**

votes

**1**answer

492 views

### Is there a generalization of Sobolev spaces for certain locally compact groups?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square ...

**7**

votes

**2**answers

460 views

### Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...

**6**

votes

**4**answers

664 views

### Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by
\begin{equation} ...

**6**

votes

**1**answer

230 views

### Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...

**6**

votes

**1**answer

188 views

### Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...

**6**

votes

**1**answer

319 views

### Compactness of Sobolev embedding for domains of finite measure

Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero ...

**6**

votes

**1**answer

154 views

### $W^{1,1}$ simplicial approximation

Let $f$ be a continuous real-valued function defined on an $n$ dimesional simplex $\Sigma\subset \mathbb{R}^n $. The classical simplicial approximation scheme provides a sequence $f_k$ of piecewise ...

**6**

votes

**0**answers

132 views

### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**6**

votes

**0**answers

78 views

### Best constant for a trace inequality

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality
$$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla ...

**5**

votes

**1**answer

420 views

### Compactness in Sobolev spaces

I was wondering whether the set $\lbrace f\in H_0^1(\Omega)|\|f\|_{L^\infty(\Omega)}\leq 1\rbrace$ is compact in $H_0^1(\Omega)$ or not. Here $\Omega$ is a convex domain in $\mathbb{R}^3$ with ...

**5**

votes

**3**answers

250 views

### A space of distributions vanishing on the boundary

The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...

**5**

votes

**2**answers

1k views

### Sobolev imbedding on Riemannian manifolds

Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.
Let ...

**5**

votes

**2**answers

264 views

### Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...

**5**

votes

**1**answer

293 views

### Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions

This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f ...

**5**

votes

**1**answer

449 views

### Books about Capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...

**5**

votes

**2**answers

327 views

### A question on density of Lipschitz functions in weighted Sobolev spaces

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$.
Let now ...

**5**

votes

**1**answer

256 views

### Embeddings of Sobolev spaces

Let $s_1,s_2\in \mathbb R$ such that $-\frac12<s_1\le s_2$.
There exists $C>0$ such that for all smooth functions $w$ , for all $r>0$,
$$\operatorname{supp} w \subset(-r,r)\Longrightarrow
...

**5**

votes

**1**answer

319 views

### Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$).
Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...

**5**

votes

**0**answers

272 views

### Which functions can be approximated by piecewise constant functions?

Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in ...

**5**

votes

**0**answers

179 views

### Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: ...

**5**

votes

**0**answers

173 views

### A finely open set, not open up to polar set?

I already asked this on M.SE, but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...

**4**

votes

**4**answers

684 views

### Variation on the Sobolev space $H^1_0$

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\},
$$
and let $C^1_c(\Omega)$ be the space of ...

**4**

votes

**1**answer

300 views

### Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
De Giorgi paper
At page ...

**4**

votes

**2**answers

846 views

### Example for the Sobolev embedding theorem when p=n.

Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then
$u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 ...

**4**

votes

**2**answers

221 views

### Finding an optimal $p$ such that $u \in L^p$

We have an $L^2$ function $u$ defined on $\mathbb{R^2}$ with compact support such that $u \in H^{2/3}$ (H stands for Sobolev spaces, as always), $\partial_y u \in L^2$, and $(x\partial_y - ...

**4**

votes

**1**answer

182 views

### Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case.
Question:
Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...

**4**

votes

**1**answer

255 views

### Equivalent Norms for the Dual of Sobolev / Bessel Spaces

Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb ...

**4**

votes

**2**answers

318 views

### Negative real order Sobolev spaces: density and representation

First, I give my motivation to ask this question. The generalised Neumann trace can be defined as
$$
{}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial ...

**4**

votes

**1**answer

286 views

### about smoothing pseudodifferential operators

Hi,
I have a question which involves pdo.
Let us consider a pseudodifferential operator $A:S(\mathbb{R}^d)\rightarrow S(\mathbb{R}^d) $ whose symbol $a(x,\xi)$ lives in the $S_{0,0}^0$ class :
$$ ...

**4**

votes

**1**answer

221 views

### projection of sobolev spaces onto cones

Consider the Sobolev space $W^{k,p}(\Omega)$ for $k\in \mathbb N$, $p\in [1,\infty]$ and some open domain $\Omega\subset \mathbb R^n$ $^*$. Then it is known that $W^{k,p}(\Omega)$ is an ordered Banach ...

**4**

votes

**2**answers

203 views

### Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ...