# 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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### fractional heat equation and spectral method

I want to apply a spectral method for the weak formulation of the equation $(-\Delta)^su=f$ $s>0$ with zero Dirichlet boundary conditions, where $(-\Delta)^s$ shall be the fractional laplacian on ...
• 69
1 vote
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### For $f \in H^s$, then $\exists g \in C_c$ such that $f=g$ a.e.

Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous ...
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• 69
1 vote
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• 69
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• 2,130
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### Norm on $H^1/2(\partial\Omega)$ by first isomorphism theorem.

I want to show that $\operatorname{ran}(T)$ equipped with \begin{equation*} \lVert v\rVert:=\inf\{u\in H^1(\Omega):Tu=v\} \end{equation*} is a Hilbert-Space. I have seen that the definition of the ...
• 41
383 views

### Fourier multipliers on $L^2(\mu)$

On $L^2(\mathbb{R}^d)$, we have $T_m$ defined $\widehat{T_m f} = m \widehat{f}$ is a bounded operator on $L^2$ if and only if $m \in L^\infty$. What can be said about the same problem for more general ...
• 460
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### Is $H(div; \Omega ) \cap H(curl; \Omega )$ compactly embedd in $L^2(\Omega)$?

I know a similar question says that $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$, which was shown in Lemma~ 2.5 of the book "Finite Elements Methods for Navier-Stokes Equations" by ...
• 69
51 views

### $H^{2}_{0}(\Omega)$ is embedded in $L^{\infty}(\Omega)$?

In a research paper https://hal.science/hal-02891557/, the authors used the embedding of $H^{2}_{0}(\Omega)$ in $L^{\infty}(\Omega)$ (see page 16-line 1) to obtain some estimates in their research ...
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### Relationship between Schwartz space and fractional Sobolev space

Let $u$ be an element of the Schwartz space $\mathcal{S}(\mathbb{R}^N)$. Given the fractional Laplacian operator $(-\Delta)^s:\mathcal{S}(\mathbb{R}^N) \to L^2(\Omega)$, with $s \in (0,1)$ , defined ...
• 745
1 vote
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### If $u\in D^{s, p}(\mathbb R^n)$, is that true that $u\in C(\mathbb R^n)$?

Let $0<s<1$, $p>1$ and $n>sp$. For $u\in C_0^{\infty}(\mathbb R^n))$ let $$[u]_{s, p} =\left(\iint_{\mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\right)^{\frac1p}$$ be the ussual ...
• 3,386
1 vote
95 views

### maximum principle for compact manifolds

Are the maximum/minimum principles on $\mathbb{R}^n$ available for both local or nonlocal operators can be modified for a compact Riemannian manifold? For instance, we have a nonlocal maximum ...
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• 745