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So we have the whole set of theory for Sobolev spaces \begin{equation} H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\}, \end{equation} and we know that they are the same as \begin{equation} W^{s,2}=\{u:D^{\alpha}u\in\mathcal{L}^2(\mathbb{R}^d)\text{ for all} |\alpha|\le s\} \end{equation} when $s\in\mathbb{N}$.

We also know that $H_s$ is useful when $s$ is a negative integer since it can be identified as the dual space of $H_{-s}$.

But what is the use of $H_s$ when $s$ is not an integer?

Thanks!

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We may have embeddings with spaces of Hölder continuous functions. –  Davide Giraudo Nov 13 '12 at 15:00
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The fractional Sobolev spaces are important when formulating elliptic boundary value problems in Sobolev spaces.

Consider the Dirichlet problem for the Poisson equation. That is, you are interested in a solution $u$ to $\Delta u = g$ on a bounded open domain $\Omega \subset \mathbb{R}^n$ that satisfies a given boundary condition $u | \partial \Omega = f$ for some $f$ that is defined on $\partial{\Omega}$.

How do you formulate "boundary" conditions when functions in Sobolev spaces aren't necessarily even continuous? The boundary $\partial \Omega$ of (a sufficiently nice open subset) $\Omega$ is of measure zero and functions in $H^k(\Omega)$ are defined a priori only a.e, and changing them on a measure zero subset doesn't affect them.

Still, if $\partial \Omega$ is nice enough (say, a $C^k$ manifold), and if $k \geq 1$ is an integer, one can show that there are trace operators $T : H^k(\Omega) \rightarrow L^2(\partial \Omega)$, that extend continuously the usual restriction map $f \mapsto f| \partial \Omega$ on $C^{\infty}(\Omega) \cap H^k(\Omega)$. What is the image of $T$? That is, what are all the possible "boundary values" of a function in $H^k(\Omega)$? It turns out to be precisely $H^{k-\frac{1}{2}}(\partial \Omega)$!

Using this, one can formulate and prove uniqueness and existence results for solutions of PDEs in Sobolev spaces. For example, one has that the map $$ u \mapsto (\Delta u, u|_{\partial \Omega}) = (\Delta u, Tu) $$ is an isomorphism of $H^1(\Omega) \rightarrow H^{-1}(\Omega) \times H^{\frac{1}{2}}(\Omega)$, and so, given $g \in H^{-1}(\Omega)$ and $f \in H^{\frac{1}{2}}(\Omega)$, the Dirichlet problem for the Poisson equation $\Delta u = g$, $u|_{\partial \Omega} = f$ has a unique solution in $H^1(\Omega)$.

There are also higher order trace operators, which correspond to normal derivatives of various orders, whose image again lies in fractional Sobolev spaces.

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