Contents

Idea

If $1\le p<\mathrm{\infty }$ 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

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

For $1\le p\le \mathrm{\infty }$, and $k\ge 1$ the Sobolev space $W_p^k=W_p^k(\Omega )$ or $W^{k,p}(\Omega )$ is the space of measurable functions $f$ on $\Omega$ such that its generalized partial derivatives ${\partial }_1^{i_1}\dots {\partial }_n^{i_n}f$ (e.g. in the sense of generalized functions) for all multiindices $i=(i_1,\dots ,i_n)\in {\mathbb{Z}}_{\ge 0}^n$ with $i_1+\dots +i_n\le k$ are in $L_p(\Omega )$. The most important case is the case of the Sobolev spaces $H^k(\Omega ):=W_2^k(\Omega )$. 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