nLab Sobolev space

Idea

If $1\leq p \lt \infty$ and $\Omega$ is a domain (in a $n$-dimensional real space with easy generalization to manifolds), one first considers the Lebesgue spaces $L_p = L_p(\Omega)$ (wikipedia) of (equivalence classes of) measurable (complex- or real-valued) functions $f$ whose (absolute values of) $p$-th powers are Lebesgue integrable; i.e. whose norm

$$\| f\|_{L_p} = \left(\int_\Omega |f|^p d\mu\right)^{1/p}$$

is finite. For $p = \infty$, one looks at the essential supremum norm $\|f\|_{L_\infty}$ instead.

For $1\leq p \leq \infty$, and $k\geq 1$ the Sobolev space $W^k_p = W^k_p(\Omega)$ or $W^{k,p}(\Omega)$ is the Banach space of measurable functions $f$ on $\Omega$ such that its generalized partial derivatives $\partial_1^{i_1}\ldots\partial_n^{i_n} f$ (e.g. in the sense of generalized functions) for all multiindices $i = (i_1,\ldots, i_n)\in\mathbb{Z}^n_{\geq 0}$ with $i_1+\ldots +i_n\leq k$ are in $L_p(\Omega)$. The most important case is the case of the Sobolev spaces $H^k(\Omega) := W^k_2(\Omega)$. Sobolev spaces are particularly important in the theory of partial differential equations.

References

• L. C. Evans, Partial Differential Equations, Amer. Math. Soc. 1998.

• R.A. Adams, Sobolev spaces, Acad. Press 1975.

• wikipedia: Sobolev space.

• Springer Encyclopaedia of Mathematics: Sobolev space

• G. Wilkin, Sobolev spaces on Euclidean space, Liber Mathematicae 2011, link

category: analysis