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Sobolev spaces

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Sobolev spaces

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous functions and with the derivatives understood in the classical sense.

Motivation

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class C1 — see smooth function). Differentiable functions are important in many areas, and in particular for differential equations. On the other hand, quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms, rather than the uniform norm. A typical example is measuring the energy of a temperature or velocity distribution by an L2-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

The integration by parts formula yields that for every uCk(Ω), where k is a natural number and for all infinitely differentiable functions with compact support φCc(Ω),

\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 ∈ Ck(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 Dαu := v.

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

Sobolev spaces with integer k

Definition

The Sobolev space Wk,p(Ω) is defined to be the set of all functions uLp(Ω) such that for every multi-index α with |α| ≤ k, the weak partial derivative Dαu belongs to Lp(Ω), i.e.

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

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

There are several choices for a norm for Wk,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)^{\frac{1}{p}}, & 1 \leq p < + \infty; \\ \max_{| \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, Wk,p(Ω) is a Banach space. For finite p, Wk,p(Ω) is also a separable space. It is conventional to denote Wk,2(Ω) by Hk(Ω) for it is a Hilbert space with the norm \| \cdot \|_{W^{k, 2}(\Omega)} .[1]

Approximation by smooth functions

Many of the properties of the Sobolev spaces cannot be seen directly from the definition. It is therefore interesting to investigate under which conditions a function uWk,p(Ω) can be approximated by smooth functions. If p is finite and Ω is bounded with Lipschitz boundary, then for any uWk,p(Ω) there exists an approximating sequence of functions umC(Ω), smooth up to the boundary such that:[2]

\left \| u_m - u \right \|_{W^{k,p}(\Omega)} \to 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[3][4]) that the space Wk,p(ℝn) can equivalently be defined as

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

with the norm

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

are called Bessel potential spaces[5] (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case p = 2.

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

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

Again, Hs,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 Wk,p(Ω) = Hk,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform Ck-boundary, k a natural number and 1 < p < ∞. By the embeddings

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

the Bessel potential spaces Hs,p(ℝn) form a continuous scale between the Sobolev spaces Wk,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

\left [ W^{k,p}(\mathbb{R}^n), W^{k+1,p}(\mathbb{R}^n) \right ]_\theta = H^{s,p}(\mathbb{R}^n),

where:

1 \leq p \leq \infty, \ 0 < \theta < 1, \ s= (1-\theta)k + \theta (k+1)= k+\theta.

Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting.[6] For an open subset Ω of ℝn, 1 ≤ p < ∞, θ ∈ (0,1) and fLp(Ω), the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by

[f]_{\theta, p, \Omega} :=\left(\int_{\Omega} \int_{\Omega} \frac{|f(x)-f(y)|^p}{|x-y|^{\theta p + n}} \; dx \; dy\right)^{\frac{1}{p}} .

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[7] Ws,p(Ω) is defined as

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

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} .

If the open subset Ω is suitably regular in the sense that there exist certain extension operators, then also the Sobolev–Slobodeckij spaces form a scale of Banach spaces, i.e. one has the continuous injections or embeddings

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

There are examples of irregular Ω such that W1,p(Ω) is not even a vector subspace of Ws,p(Ω) for 0 < s < 1.

From an abstract point of view, the spaces Ws,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) = \left (W^{k,p}(\Omega), W^{k+1,p}(\Omega) \right)_{\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 special cases of Besov spaces.[4]

Traces

Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If u ∈ C(Ω), those boundary values are described by the restriction u|_{\partial\Omega}. However, it is not clear how to describe values at the boundary for u ∈ Wk,p(Ω), as the n-dimensional measure of the boundary is zero. The following theorem[2] resolves the problem:

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

Tu &= u|_{\partial\Omega} && u\in W^{1,p}(\Omega)\cap C(\overline{\Omega}) \\ \left\|Tu\right\|_{L^p(\partial\Omega)}&\leq c(p,\Omega)\|u\|_{W^{1,p}(\Omega)} && u\in W^{1,p}(\Omega). \end{align}

Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W1,p(Ω) for well-behaved Ω. Note that the trace operator T is in general not surjective, but for 1 < p < ∞ it maps onto the Sobolev-Slobodeckij space W^{1-\frac{1}{p},p}(\partial\Omega).

Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality

W_0^{1,p}(\Omega)= \left \{u\in W^{1,p}(\Omega): Tu=0 \right \},

where

W_0^{1,p}(\Omega):= \left \{u\in W^{1,p}(\Omega): \exists \{u_m\}_{m=1}^\infty\subset C_c^\infty(\Omega), \ \textrm{such} \ \textrm{that} \ u_m\to u \ \textrm{in} \ W^{1,p}(\Omega) \right \}.

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in W1,p(Ω) can be approximated by smooth functions with compact support.

Extensions

For a function f ∈ Lp(Ω) on an open subset Ω of ℝn, its extension by zero

Ef := \begin{cases} f & \textrm{on} \ \Omega, \\ 0 & \textrm{otherwise} \end{cases}

is an element of Lp(ℝn). Furthermore,

\left\| Ef\right\|_{L^p(\mathbb{R}^n)}=\left\| f\right\|_{L^p(\Omega)}.

In the case of the Sobolev space W1,p(Ω) for 1 ≤ p ≤ ∞ , extending a function u by zero will not necessarily yield an element of W1,p(ℝn). But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C1), then for any bounded open set O such that Ω⊂⊂O (i.e. Ω is compactly contained in O), there exists a bounded linear operator[2]

E: W^{1,p}(\Omega)\rightarrow W^{1,p}(\mathbb{R}^n),

such that for each u ∈ W1,p(Ω): Eu = u a.e. on Ω, Eu has compact support within O, and there exists a constant C depending only on p, Ω, O and the dimension n, such that

\left\| Eu\right\|_{W^{1,p}(\mathbb{R}^n)}\leq C\left\|u\right\|_{W^{1,p}(\Omega)}.

We call Eu an extension of u to ℝn.

Sobolev embeddings

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives or large p result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.

Notes

References

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External links

  • Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci (2011). "Hitchhiker's guide to the fractional Sobolev spaces".
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