The Sobolev Space $H^{1/2}$

In my course on linear PDEs, the professor used $H^{1/2}$ without defining it, and I have been looking on google trying to find a definition, but the only related thing I found was $H^{-1/2}$ as being the dual space to $H^{1/2}$ which does not really help. Plugging in the one half in the defintion of the standard Sobolev spaces $H^m$ does not make any sense. Could someone quickly help me out there?

Thank you.

Solution 1:

There are multiple definitions of $H^{1/2}(\partial Ω)$ which are equivalent if the boundary is regular enough (Lipschitz continuous). The technically simplest, and how it usually appears in lectures on weak solutions for partial differential equations, is as the range of the trace operator $tr\colon H^1(Ω) \to L^2(\partial Ω)$:

$$ \begin{align}H^{1/2}(\partial Ω) &:= tr(H^1(\Omega)) := \{ v \in L^2(\partial Ω) \;|\; \exists u \in H^1(Ω)\colon tr(u) =v \},\\ \| v \|_{H^{1/2}(\partial Ω)} &:= \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; \tilde u \in H^1(Ω) \land tr(\tilde u) = v \}.\end{align}$$

The definition of the norm arises as follows. By the First isomorphism theorem for Banach Spaces, the trace operator induces an isomorphism $$ \begin{align}\widehat{tr}\colon H^1(Ω) / \operatorname{ker} tr &\to tr(H^1(\Omega)), \\ [u] &\mapsto tr(u)\end{align} $$ where $\operatorname{ker} tr$ is the kernel of the trace operator, $[u] \in H^1(Ω) / \operatorname{ker} tr$ denotes an equivalence class with representative $u \in H^1(Ω)$ and the norm on the quotient space is given by $$ \| [u] \|_{H^1(Ω) / \operatorname{ker} tr} := \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; \tilde u \in [u] \}. $$ This is a general construct for the quotient norm on Banach spaces. As a side remark, there holds $\operatorname{ker} tr = H^1_0(Ω)$ (the latter space is defined as completion of $C^\infty_0(\Omega)$ in $H^1(\Omega)$). One can then define a norm on $H^{1/2}(\partial Ω)$ using $\widehat{tr}$:

$$ \| v \|_{H^{1/2}(\partial Ω)} := \| \widehat{tr}^{-1}(v) \|_{H^1(Ω) / \operatorname{ker} tr} = \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; \tilde u \in \widehat{tr}^{-1}(v) \}$$

using that $\tilde u \in \widehat{tr}^{-1}(v)$ if and only if $\tilde u \in H^1(\Omega)$ and $tr(\tilde u) = v$ one arrives at the expression of the norm given in the beginning.

This definition of $H^{1/2}(\partial Ω)$ is not very useful if one wishes to check whether a specific function $v \in L^2(\partial Ω)$ is in $H^{1/2}(\partial Ω)$ and it does not explain the name $H^{1/2}(\partial Ω)$ (which came later historically).

The other definition of $H^{1/2}(\partial Ω)$ I present here is quite technical in detail as $\partial Ω$ is a $(n-1)$-dimensional manifold. In case $\partial Ω$ is a plane you have $\partial Ω \cong \mathbb R^{n-1}$ and you end up having to define $H^{1/2}(Ω')$ for $Ω' \subset \mathbb R^{n-1}$. For a general Lipschitz boundary you can "straighten" your boundary locally to look like a plane (this is a general technique while working with manifolds) and in the end you ask for a transformation of your boundary function to be in $H^{1/2}(Ω')$. (See [1] for details.)

All in all, you end up having to define $H^{1/2}(Ω')$. There are multiple ways for doing that, one using the Hölder-like seminorms as mentioned by Thomás, one using the Fourier coefficients (see Fractional Sobolev Spaces on Wikipedia) and one using interpolation between $L^2(Ω')$ and $H^1(Ω')$ (see [1] again).

For understanding the actual behavior of functions in $H^{1/2}(Ω')$ the definition using the Hölder-like norm (Sobolev-Slobodeckij norm) is probably the best: $$H^{1/2}(Ω') = \left\{ v ∈ L^2(Ω') \;|\; \| v \|_{L^2(Ω')} + \int_{Ω'}\int_{Ω'}\frac{|v(x)-v(y)|^2}{|x-y|^{n+1}} dx \, dy < \infty \right\}$$ Note that the additional integral term is somewhat like a Hölder condition. I like to think of $H^1(Ω) \subset H^{1/2}(Ω) \subset L^2(Ω)$ as something analogous to $C^1(Ω) \subset C^{1/2}(Ω) \subset C^0(Ω)$ in terms of regularity. That this is really analogous can be made precise using interpolation theory, which allows one to define spaces $H^s(Ω)$ for any $0 < s < 1$ "in-between" $L^2(\Omega)$ and $H^1(Ω)$, where the trace space appears as special case for $s = 1/2$.

The only source claiming the equivalence of the norms I know of is [2], but my Italian is not sufficient to follow the argument.

[1] Lions, J. L., & Magenes, E. (1972). Non-Homogeneous Boundary Value Problems and Applications.

[2] Gagliardo, E. (1957). Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rendiconti Del Seminario Matematico Della Università Di Padova, 27, 284–305.