Fractional Sobolev Spaces and Trace Theory

Question

I've been working with fractional Sobolev Spaces for a while and I still don't get how is it connected to trace theory, is there any literature which goes deeper into such relationship?

From the boook

Fractional Spaces for the Theory of Elliptic PDE by Françoise Demengel Gilbert Demengel

It says that the need of such spaces lies on the existence of the trace for the derivatives, which makes sense since we have things like Neumman conditions. However it doesn't really tell you how a trace is defined for derivatives.

The big question is why on such spaces, what is the real advantage on fractional Sobolev spaces and the relation to the distance of traces?

And if there is any intuitive idea of such spaces and the need of them?

Thanks in advance.

Answer 1

Actually I would say the first thing to remark, is that if a function is in $L^p$ then it is only defined almost everywhere. Therefore, you just cannot in general define its trace since it would mean to get the values of the function on a set of measure $0$ (since of dimension smaller). However, if the function is continuous, you see that you can easily define the trace of your function and it will be continuous.

From this preliminary analysis, you deduce that in general, you need some regularity assumptions to define the trace of a function.

Now look at a function with a local singularity such as $$ f(x) = \frac{1}{|x|^a} $$ You can see that this function is locally in $L^p(\mathbb{R}^d)$ if $p<d/a$, but if you take the trace on a set of smaller dimension and containing $0$, you see that the trace will only locally be in $L^q$ with $q<d/a - 1/a$, so you loose a part of the integrability when you take the trace. This is from my point of view a way to understand intuitively why starting from a function with a certain regularity, you loose a part of the regularity when taking the trace.

The fractional Sobolev spaces created by real interpolation were investigated a lot by Jacques-Louis Lions, and actually were sometimes called trace spaces. A good reference is the book by Luc Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Chapter 16 treat the case of the $L^2$ based $H^s$ Sobolev spaces and Chapter 40 of the more general case of $L^p$ based Sobolev spaces $W^{s,p}$.

An interesting part is also Chapter 33 about the space $H^\frac{1}{2}_{00}$, which in some sense the critical case where one can still define a trace on the border (since $H^s_0(\Omega) = H^s(\Omega)$ when $s\leq 1/2$).

Answer 2

Fractional Sobolev spaces appear naturally as the correct range of trace maps. Let me explain this for $L^2$-based Sobolev spaces on a smooth, bounded domain $\Omega \subset \mathbb{R}^d$. In this case one has a continuous trace map $$ \tau:H^s(\Omega)\rightarrow L^2(\partial \Omega), \quad s> 1/2, $$ that extends restriction $u\mapsto u \vert_{\partial \Omega}$ from $H^s(\Omega)\cap C(\Omega)$ (where it is well defined) to the whole space. (For now you may view $s$ as an integer, it will only become necessary to pass to non-integers in a bit.) A natural question is then to ask which functions in $L^2(\partial \Omega)$ can be extended to an $H^s$-function in $\Omega$. The answer is $$ \mathrm{range}(\tau)=H^{s-1/2}(\partial \Omega), $$ i.e. a fractional Sobolev space pops up, even if you started with $s\in \mathbb{N}$. In fact, the trace map is even continuous when considered as operator $$ \tau:H^s(\Omega)\rightarrow H^{s-1/2}(\partial \Omega). $$ Note that this is a stronger continuity statement, as $H^{s-1/2}$ carries a finer topology than $L^2$. Further, there is a continuous extension map $$ E: H^{s-1/2}(\partial \Omega) \rightarrow H^s(\Omega), \quad E \circ \tau = \mathrm{id}, $$ which is useful when considering boundary value problems with non-smooth boundary data (e.g. you can only hope for a solution in $H^s$ when your boundary data lies in $H^{s-1/2}$.) This all can be neatly summarised by saying that $$ 0\rightarrow H^s_0(\Omega) \hookrightarrow H^s(\Omega) \xrightarrow{\tau} H^{s-1/2}(\partial \Omega) \rightarrow 0 $$ is a (split) exact sequence of Hilbert spaces.

All of this is explained in chapter 4 of Taylor's book 'Partial Differential Equaitons I: Basic Theory'.

Answer 3

$\newcommand{\ext}{\operatorname{ext}}$ Don't know how directly this is related to the OP, but, it won't hurt.

Lemma: Let $M$ be a smooth closed $n$-dimensional Riemannian manifolds with boundary, $n\geq 2$ and let $1<p<\infty$ There is a unique bounded linear trace operator $$ \operatorname{Tr}:W^{1,p}(M)\to W^{1-\frac{1}{p},p}(\partial M) $$ such that $\operatorname{Tr}f=f|_{\partial M}$ for functions $f\in C^\infty(M)$ that are smooth up to the boundary. Moreover there is a bounded linear extension operator $$ \ext_{\partial M}:W^{1-\frac{1}{p},p}(\partial M)\to W^{1,p}(M) $$ such that $\operatorname{Tr}\circ\ext_{\partial M}=\operatorname{Id}$ on the space $W^{1-\frac{1}{p},p}(\partial M)$.

Therefore, fractional Sobolev spaces are the image of the trace operator $ Tr: W^{1,p}(M) \to L^p(\partial) $. We did know from classical Sobolev theory that such a trace exists but fractional Sobolev spaces characterie ALL $L^p$ functions on the boundary that arise as traces of $W^{1,p}$'s.