Tag Archives: weak derivatives

Fractional Sobolev Spaces

Recall for a function f\in L^1(\Omega) for \Omega\subseteq\mathbb{R}^n we define the Fourier transform of f by

\displaystyle\hat{f}(x)=F(f)(x)=\int_\Omega e^{-2\pi ix\cdot\xi}f(\xi)\,d\xi

and the inverse Fourier transform by

\displaystyle \check{f}(x)=F^{-1}(f)(x)=\int_\Omega e^{2\pi i x\cdot\xi}f(\xi)\,d\xi.

If f,\hat{f}\in L^1(\Omega), then \check{\hat{f}}=\hat{\check{f}}=f a.e..

Now if f\in L^1 and D^\alpha (f)\in L^1 with \alpha\in\mathbb{Z}^n, we say D^\alpha f is the weak derivative of f provided

\displaystyle\int_\Omega f D^\alpha\phi\,dx=(-1)^{|\alpha|}\int_\Omega D^\alpha(f)\phi\,dx

with \phi\in C_C^\infty(\Omega), |\alpha|=\sum_i\alpha_i, and

\displaystyle D^\alpha\phi=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}\phi.

The Sobolev space W^{k,p}(\Omega) is then defined as the set of all functions f\in L^p(\Omega) such that D^\alpha(f)\in L^p(\Omega) for all |\alpha|\leq k.  These become Banach spaces under the norm

\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\int_\Omega \left|D^\alpha(f)\right|^p\,dx\right)^{1/p}

and Hilbert when p=2, whence we denote H^k(\Omega)=W^{k,2}(\Omega).

For 1<p<\infty we can say that W^{k,p}(\mathbb{R}^n) consists of f\in L^p(\mathbb{R}^n) such that

\displaystyle F^{-1}\circ(1+|2\pi x|^2)^{k/2}F\circ f=(1-\Delta)^{k/2}f(x)\in L^p(\mathbb{R}^n)

(using Fourier multipliers).

Bessel Potential Spaces:

We can generalize with r\in\mathbb{R} and 1<p<\infty and define the Bessel potential space W^{r,p}(\mathbb{R}^n) as all f\in L^p such that \|f\|_{r,p}:=\|(1-\Delta)^{r/2}f\|_p<\infty.  Note the fractional power of the Laplacian is defined by virtue of the fact that L^p\cap L^1 is dense in L^p, so we say

\displaystyle(1-\Delta)^{r/2}(f):=(F^{-1}\circ (1+|2\pi x|^2)\circ F)(f).

 These spaces satisfy our desire of being Banach (and Hilbert when p=2).

Sobolev-Slobodeckij Spaces:

There is an alternative approach.  Recall that the Holder space C^{k,\gamma}(\Omega) is defined as all functions f\in C^k(\Omega) such that

\displaystyle\|f\|_{C^{k,\gamma}}=\sum_{|\alpha|\leq k}\sup_{x\in\Omega}|D^\alpha|+\sum_{|\alpha|=k}\sup_{x,y\in\Omega}\frac{|D^\alpha f(x)-D^\alpha f(y)|}{|x-y|^\gamma}<\infty.

That is, it is the set of functions on \Omega which are C^k and whose k-th partial derivatives are bounded and Holder continuous of degree \gamma.  These spaces are Banach under the above norm.  We can generalize the Sobolev spaces to incorporate similar properties.  Let us define the Slobodeckij norm for f\in L^p(\Omega) with 1\leq p<\infty and \theta\in (0,1) by

\displaystyle [f]_{\theta,p}=\int_\Omega\int_\Omega\frac{|f(x)-f(y)|^p}{|x-y|^{\theta p+n}}\,dxdy.

The corresponding Sobolev-Slobodeckij space W^{s,p}(\Omega) is defined as all functions f\in W^{\lfloor s\rfloor,p}(\Omega) such that

\displaystyle\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}<\infty

where \theta=s-\lfloor s\rfloor\in(0,1).  This becomes a Banach space under the norm

\displaystyle\|f\|_{s,p}=\|f\|_{W^{\lfloor s\rfloor,p}}+\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}.

[1] http://en.wikipedia.org/wiki/Sobolev_space (unclear text references)

[2]  Lieb, Elliot and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

[3]  Evans, Lawrence.  Partial Differential Equations.  Graduate Studies in Mathematics.  Vol. 19.  American Mathematical Society.  1998.

Distributions

Let \Omega\subseteq\mathbb{R}^n and C_c^\infty(\Omega) denote the set of smooth functions on \Omega with compact support.

Definition 1.  Let \alpha\in\mathbb{N}^n and \{\phi_k\} be a sequence in C_c^\infty(\Omega).  We say \{\phi_k\} converges to \phi\in C_c^\infty(\Omega), denoted \phi_k\to\phi, if

  1. \mbox{supp~}\phi_k-\phi\subseteq K for all k for a fixed compact K\subseteq\Omega,
  2. D^\alpha\phi_m\to D^\alpha\phi uniformly on K.

By D^\alpha\phi we mean

\displaystyle\frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}}\cdots\frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}}\,\phi.

C_c^\infty(\Omega) is a vector space, and the above notion of convergence allows us to define closed sets and hence turns C_c^\infty(\Omega) into a topological vector space called the space of test functions of \Omega, and we denote it D(\Omega).

Definition 2.  A distribution is a continuous linear functional on D(\Omega).  We will denote the set of distributions on \Omega by D(\Omega)^*–noting that we are only considering continuous linear functionals.

We say a sequence of distributions \{T_k\} converges to the distribution T, denoted T_k\to T, if T_k(\phi)\to T(\phi) for all \phi\in D(\Omega).  This gives us a topological vector space structure on D(\Omega)^* as well.

Definition 3.  Let T be a distribution and \alpha\in\mathbb{N}^n.  We define the \alpha-th distributional (weak) derivative of T by

D^\alpha(T)(\phi)=(-1)^{|\alpha|}T(D^\alpha\phi)

where |\alpha|=\sum\alpha_i.  We may also write

\displaystyle\partial_i^kT=(-1)^kT\left(\frac{\partial^k}{\partial x_i^k}\phi\right).

We will omit k if k=1.  The gradient of T is simply

\nabla T=(\partial_1T,...,\partial_nT).

Let us define the space L_{loc}^p(\Omega) as the space of functions f:\Omega\to\mathbb{R} such that

\displaystyle\|f\|_{L^p(K)}=\left(\int_K|f|^p\,dx\right)^{1/p}<\infty

for all compact K\subseteq\Omega.  We define strong convergence on L_{loc}^p(\Omega) as convergence in L^p(\Omega) and weak convergence on L_{loc}^p(\Omega) as weak (pointwise) convergence in L^p(K) for all compact K\subseteq\Omega.

Let 1\leq p\leq q, then clearly

L^q(\Omega)\subseteq L_{loc}^q(\Omega)\subseteq L_{loc}^p(\Omega).

Let f\in L_{loc}^1(\Omega) and \phi\in D(\Omega) and define

\displaystyle T_f(\phi)=\int_\Omega f\phi\,dx.

This integral is finite since both functions are integrable (use integration by parts and establish bounds).  Also if \phi_k\to\phi uniformly, then

\begin{array}{lcl}|T_f(\phi)-T_f(\phi_k)|&=&\displaystyle\left|\int_\Omega(\phi(x)-\phi_k(x))f(x)\,dx\right|\\&\leq&\displaystyle\sup_{x\in K}|\phi(x)-\phi_k(x)|\int_K|f(x)|\,dx\\&<&\infty\end{array}

where K=\mbox{supp~}(\phi-\phi_k), which gives us the continuity of T_f.  So T_f\in D(\Omega)^*.  We also have

\displaystyle (D^\alpha T_f)(\phi):=(-1)^{|\alpha|}\int_\Omega(D^\alpha\phi)f\,dx=\int_\Omega(D^\alpha f)\phi\,dx=T_{D^\alpha f}(\phi).

[1]  Lieb, Elliott and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.