Sobolev space - Imported Knowlpedia Content

Definition

Let $\Omega\subseteq\mathbb R^n$ be open, let $k$ be a nonnegative integer, and let $1\leq p\leq\infty$ . The Sobolev space $W^{k,p}(\Omega)$ consists of equivalence classes of functions $u\in L^p(\Omega)$ whose weak derivatives $D^\alpha u$ belong to $L^p(\Omega)$ for every multi-index $\alpha$ with $|\alpha|\leq k$ . For $p<\infty$ , its standard norm is

\lVert u\rVert_{W^{k,p}} =\left(\sum_{|\alpha|\leq k}\lVert D^\alpha u\rVert_{L^p}^p\right)^{1/p};

for $p=\infty$ , use the maximum of the essential-supremum norms. This norm makes $W^{k,p}(\Omega)$ a Banach space.

Hilbert and fractional cases

When $p=2$ , $W^{k,2}(\Omega)$ is a Hilbert space and is commonly denoted $H^k(\Omega)$ . For noninteger $s$ , the notation $H^s$ usually denotes a fractional Sobolev space defined by Fourier multipliers, interpolation, or local charts. On sufficiently regular domains these constructions agree in their appropriate ranges, but boundary behavior and quotient-versus- restriction definitions must be specified Adams--Fournier, Chapters 3 and 7.

Completeness, density, and traces

Weak differentiation is closed under $L^p$ -convergence, which underlies completeness. On an arbitrary open set, $C_c^\infty(\Omega)$ is dense in $W^{k,p}_0(\Omega)$ by definition, but it need not be dense in all of $W^{k,p}(\Omega)$ . On domains with suitable boundary regularity, trace theorems assign boundary values to Sobolev functions even though pointwise restriction is not defined for a general $L^p$ -class.

Examples and non-examples

The function $u(x)=|x|$ lies in $W^{1,\infty}((-1,1))$ , with weak derivative $\operatorname{sgn}(x)$ , although it is not classically differentiable at zero. A jump discontinuity on an interval lies in $L^p$ but not in $W^{1,p}$ for $p\geq1$ , because its distributional derivative contains a point mass rather than an $L^p$ -function.

Conventions and scope

References

Robert A. Adams and John J. F. Fournier, Sobolev Spaces, 2nd ed., Academic Press, 2003. Publisher record. Relevant: Chapters 3 and 7 on integer-order and fractional Sobolev spaces.
Lawrence C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010. DOI record. Relevant: Chapter 5 on Sobolev spaces, approximation, extensions, and traces.