# Questions tagged [fractional-sobolev-spaces]

This tag is devoted to any problem concerning fractional Sobolev spaces which are approximation of the classical Sobolev spaces

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• 301
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• 301
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### The trace theorem for functions in $H^{1/2}(\Omega)$

In my textbook, it said that we have the trace operator on Sobolev space like this: (Suppose $\Omega$ is a nice domain in $R^d$) \begin{equation*} H^{s}(\Omega) \hookrightarrow H^{s-\frac{1}{2}}(\...
2 votes
0 answers
35 views

### Inequality $(\alpha f,g)_{L^2}\leq C\|\alpha\|_{W}\|f\|_{H^{1/2}}\|g\|_{H^{3/2}}$

Let $f\in H^{1/2}(S)$, $g\in H^{3/2}(S)$, $\alpha\in L^\infty(S)$, where $S$ is a closed smooth $n$-dimensional surface (e.g, $n$-dimensional sphere). Problem: to find such Banach space $W$ that the ...
• 61
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### Why is dirac delta in $H^{-1/2-\epsilon}$ and why is multiplication not well-defined in that space?

I read on this Sobolev space with negative index question that the dirac delta distribution belongs to $H^{-1/2-\epsilon}(\mathbb{R})$. I was trying to figure out why that is the case but it wasn't ...
• 1,224
1 vote
0 answers
59 views

### Does there exist a function in $B_{\infty,1}^{1/2}\setminus H^{1/2}$?

Does there exist a function in $B_{\infty,1}^{1/2}([0,1])\setminus H^{1/2}([0,1])$? If so can it be constructed explicitly? A reference will be much appreciated.
• 1,107
2 votes
0 answers
32 views

### Sobolev embedding theorem on Manifolds

I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional Sobolev spaces. The references I know either don't deal with fractional spaces, or don't ...
• 669
2 votes
0 answers
66 views

### Does $\int_{E^c}\int_{E}\frac{dy\, dx}{|x-y|^{n+\alpha}}$ converge for any $E$?

Question: Given an $\alpha\in(0,n)$, does there exist a measurable set $E\subset\mathbb R^n$ with $\mu(E)\in(0,\infty)$ such that $$\int_{E^c}\int_{E}\frac{dy\, dx}{|x-y|^{n+\alpha}}<+\infty$$ ? (...
• 10.7k
3 votes
0 answers
65 views

### Bounding the Laplacian of a function using its boundary data

Let $U \subset \mathbb{R}^n$ be a bounded domain with smooth boundary, which consists of two smooth disjoint pieces $\partial U = \Gamma_1 \cup \Gamma_2$. Let $f \in H^2(U)$ be a given function and ...
1 vote
1 answer
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### Sobolev spaces are Hilbert spaces

Define $H^s(\mathbb{R}^n)=\lbrace u\in \mathcal{S}'(\mathbb{R}^n): (1+\vert y\vert^2)^\frac{s}{2}\hat{u}\in L^2(\mathbb{R}^n)\rbrace$ where $\mathcal{S}'(\mathbb{R}^n)$ is the space of tempered ...
1 vote
0 answers
39 views

• 1,640
4 votes
1 answer
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• 2,580
0 votes
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### Is the embedding of $H^{1+s}(\Omega)$ (fractional Sobolev space) in $H^1(\Omega)$ compact?

Here, $s\in]0, 1]$ and $\Omega$ is a bounded open connected Lip. domain in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. I know that $H^1(\Omega)$ is compact in $L^2(\Omega)$, but I have no idea how use it to ...
• 63
1 vote
2 answers
86 views

### Estimate of supremum of a function by an integral

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, $n\geq 2$. For $r>0$, denote by $B_r(x_0)=\{x\in\mathbb{R}^n:|x-x_0|<r\}$ whose closure is a proper subset of $\Omega$. Let \$u\in W^{1,p}(\...
• 409