User:FunctionspaceInvader

Motivation

The integration by parts formula yields that for every u ∈ C^k(Ω), where k is a natural number and for all infinitely differentiable functions with compact support $\varphi \in C_{c}^{\infty }(\Omega )$,

$\int _{\Omega }uD^{\alpha }\varphi \;dx=(-1)^{|\alpha |}\int _{\Omega }\varphi D^{\alpha }u\;dx$,

where α a multi-index of order |α| = k and Ω is an open subset in Rⁿ. Here, the notation

$D^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\dots \partial x_{n}^{\alpha _{n}}}},$

is used.

The left-hand side of this equation still makes sense if we only assume u to be locally integrable. If there exists a locally integrable function v, such that

$\int _{\Omega }uD^{\alpha }\varphi \;dx=(-1)^{|\alpha |}\int _{\Omega }\varphi v\;dx,\ \ \ \ \varphi \in C_{c}^{\infty }(\Omega ),$

we call v the weak α-th partial derivative of u. If there exists a weak α-th partial derivative of u, then it is uniquely defined almost everywhere. On the other hand, if u ∈ C^k(Ω), then the classical and the weak derivative coincide. Thus, if v is a weak α-th partial derivative of u, we may denote it by D^αu := v.

The Sobolev spaces W^k,p(Ω) combine the concepts of weak differentiability and Lebesgue norms.

Sobolev spaces with integer k

Definition

The Sobolev space W^k,p(Ω) is defined to be the set of all functions u ∈ L^p(Ω) such that for every multi-index α with |α| ≤ k, the weak partial derivative $D^{\alpha }u$ belongs to L^p(Ω), i.e.

$W^{k,p}(\Omega )=\{u\in L^{p}(\Omega ):D^{\alpha }u\in L^{p}(\Omega )\,\,\forall |\alpha |\leq k\}.$

Here, Ω is an open set in Rⁿ and 1 ≤ p ≤ +∞. The natural number k is called the order of the Sobolev space W^k,p(Ω).

There are several choices for a norm for W^k,p(Ω). The following two are common and are equivalent in the sense of equivalence of norms:

$\|u\|_{W^{k,p}(\Omega )}:={\begin{cases}\left(\sum _{|\alpha |\leq k}\|D^{\alpha }u\|_{L^{p}(\Omega )}^{p}\right)^{1/p},&1\leq p<+\infty ;\\\sum _{|\alpha |\leq k}\|D^{\alpha }u\|_{L^{\infty }(\Omega )},&p=+\infty ;\end{cases}}$

and

$\|u\|'_{W^{k,p}(\Omega )}:={\begin{cases}\sum _{|\alpha |\leq k}\|D^{\alpha }u\|_{L^{p}(\Omega )},&1\leq p<+\infty ;\\\sum _{|\alpha |\leq k}\|D^{\alpha }u\|_{L^{\infty }(\Omega )},&p=+\infty .\end{cases}}$

With respect to either of these norms, W^k,p(Ω) is a Banach space. For finite p, W^k,p(Ω) is also a separable space. It is conventional to denote W^k,2(Ω) by H^k(Ω) for it is a Hilbert space with the norm $\|\cdot \|_{W^{k,2}(\Omega )}$.

Approximation by smooth functions

A lot of properties of the Sobolev spaces cannot be seen directly from the definition. It is therefore interesting to investigate under which conditions a function u ∈ W^k,p(Ω) can be approximated by smooth functions. If p is finite and Ω is bounded with Lipschitz boundary, then for any u ∈ W^k,p(Ω) there exists an approximating sequence of functions $u_{m}\in C^{\infty }({\overline {\Omega }})$, smooth up to the boundary such that $\|u_{m}-u\|_{W^{k,p}(\Omega )}\rightarrow 0$.

Sobolev spaces with non-integer k

Bessel potential spaces

For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers, see ) that the space W^k,p(Rⁿ) can equivalently be defined as

$W^{k,p}(\mathbb {R} ^{n})=H^{k,p}(\mathbb {R} ^{n}):=\{f\in L^{p}(\mathbb {R} ^{n})\mid {\mathcal {F}}^{-1}(1+|\xi |^{2})^{\frac {k}{2}}{\mathcal {F}}f\in L^{p}(\mathbb {R} )^{n}\}$

with the norm

$\|f\|_{H^{k,p}(\mathbb {R} ^{n})}:=\|{\mathcal {F}}^{-1}(1+|\xi |^{2})^{\frac {k}{2}}{\mathcal {F}}f\|_{L^{p}(\mathbb {R} ^{n})}$.

This motivates Sobolev spaces with non-integer order since in the above definition we can replace k by any real number s. The resulting spaces

$H^{s,p}(\mathbb {R} ^{n}):=\{f\in L^{p}(\mathbb {R} ^{n})\mid {\mathcal {F}}^{-1}(1+|\xi |^{2})^{\frac {s}{2}}{\mathcal {F}}f\in L^{p}(\mathbb {R} )^{n}\}$

are called Bessel potential spaces and are denoted by H^s,p(Rⁿ). They are Banach spaces in general and Hilbert spaces in the special case p = 2 .

For an open set Ω ⊆ Rⁿ, H^s,p(Ω) is the set of restrictions of functions from H^s,p(Rⁿ) to Ω equipped with the norm

$\|f\|_{H^{s,p}(\Omega )}:=\inf\{\|g\|_{H^{s,p}(\mathbb {R} ^{n})}\mid g\in H^{s,p}(\mathbb {R} ^{n}),g_{|\Omega }=f\}$.

Again, H^s,p(Ω) is a Banach space and in the case p = 2 a Hilbert space.

Using extension theorems for Sobolev spaces, it can be shown that also W^k,p(Ω) = H^k,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform C^k-boundary, k a natural number and 1 < p < ∞. By the embeddings

$H^{k,p}(\mathbb {R} ^{n})\hookrightarrow H^{s,p}(\mathbb {R} ^{n})\hookrightarrow H^{s',p}(\mathbb {R} ^{n})\hookrightarrow H^{k+1,p}(\mathbb {R} ^{n}),\quad k\leq s\leq s'\leq k+1$

the Bessel potential spaces H^s,p(Rⁿ) form a continuous scale between the Sobolev spaces W^k,p(Rⁿ). From an abstract point of view, the Bessel potential spaces occur as complex interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms it holds that

${\big [}W^{k,p}(\mathbb {R} ^{n}),W^{k+1,p}(\mathbb {R} ^{n}){\big ]}_{\theta }=H^{s,p}(\mathbb {R} ^{n}),\quad 1\leq p\leq \infty ,\theta \in (0,1),s=(1-\theta )k+\theta (k+1).$

Sobolev-Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting. For an open subset Ω of Rⁿ, 1 ≤ p < ∞, θ ∈ (0,1) and f ∈ L^p(Ω), the Slobodeckij seminorm is (roughly analogous to the Hölder seminorm) defined by

$[f]_{\theta ,p,\Omega }:=\int _{\Omega }\int _{\Omega }{\frac {|f(x)-f(y)|}{|x-y|^{\theta p+n}}}\;dx\;dy$.

Let s > 0 be not an integer and set $\theta =s-\lfloor s\rfloor \in (0,1)$. Using the same idea as for the Hölder spaces, the Sobolev-Slobodeckij space W^{s, p}(Ω) is defined as

$W^{s,p}(\Omega ):=\{f\in W^{\lfloor s\rfloor ,p}(\Omega )\mid \sup _{|\alpha |=\lfloor s\rfloor }[D^{\alpha }f]_{\theta ,p,\Omega }<\infty \}$.

It is a Banach space for the norm

$\|f\|_{W^{s,p}(\Omega )}:=\|f\|_{W^{\lfloor s\rfloor ,p}(\Omega )}+\sup _{|\alpha |=\lfloor s\rfloor }[D^{\alpha }f]_{\theta ,p,\Omega }$.

The Sobolev-Slobodeckij spaces give a second continuous scale between the Sobolev spaces, i.e. one has the embeddings

$W^{k,p}(\Omega )\hookrightarrow W^{s,p}(\Omega )\hookrightarrow W^{s',p}(\Omega )\hookrightarrow W^{k+1,p}(\Omega ),\quad k\leq s\leq s'\leq k+1$.

From an abstract point of view, the spaces W^{s, p}(Ω) coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:

$W^{s,p}(\Omega )={\big (}W^{k,p}(\Omega ),W^{k+1,p}(\Omega ){\big )}_{\theta ,p},\quad k\in \mathbb {N} ,s\in (k,k+1),\theta =s-\lfloor s\rfloor$.

Sobolev-Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are a special case of the so-called Besov spaces.

Traces

Sobolev spaces are often considered when investigating partial differential equations. As boundary conditions are crucial in the concept of partial differential equations, we would like to have a possibility of speaking about boundary values of Sobolev functions. If u ∈ C(Ω), those boundary values are described by the restriction $u|_{\partial \Omega }$, which is obviously well-defined. However, it is not clear how to describe values at the boundary for u ∈ W^k,p(Ω), as the n-dimensional measure of the boundary is zero. Nevertheless, we have the following theorem:

Trace Theorem. Assume Ω is bounded with continuously differentiable boundary. Then there exists a bounded linear operator $T:W^{1,p}(\Omega )\rightarrow L^{p}(\partial \Omega )$ such that

$Tu=u|_{\partial \Omega },\ \ \ \ u\in W^{1,p}(\Omega )\cap C({\overline {\Omega }})$

and

$\left\|Tu\right\|_{L^{p}(\partial \Omega )}\leq c(p,\Omega )\left\|u\right\|_{W^{1,p}(\Omega )},\ \ \ \ u\in W^{1,p}(\Omega ).$

Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W^1,p(Ω) for well-behaved Ω. Note that the operator T is in general not surjective, but maps for p ∈ (1,∞) onto the Sobolev-Slobodeckij space W^1-1/p,p(Ω).
The functions u in W^1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized in a very simple way. Define therefore the space

$W_{0}^{1,p}(\Omega ):=\{u\in W^{1,p}(\Omega ):{\textrm {there}}\ {\textrm {exists}}\ \{u_{m}\}_{m=1}^{\infty }\subset C_{c}^{\infty }({\overline {\Omega }}),\ {\textrm {such}}\ {\textrm {that}}\ u_{m}\rightarrow u\ {\textrm {in}}\ W^{1,p}(\Omega )\}.$

Then for bounded Ω with continuously differentiable boundary it holds that

$W_{0}^{1,p}(\Omega )=\{u\in W^{1,p}(\Omega ):Tu=0\}.$

In other words, for Ω as above trace-zero functions in W^1,p(Ω) can be approximated by smooth functions with compact support.