nLab
Sobolev space

Contents

Idea

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

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

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

For 1p1\leq p \leq \infty, and k1k\geq 1 the Sobolev space W p k=W p k(Ω)W^k_p = W^k_p(\Omega) or W k,p(Ω)W^{k,p}(\Omega) is the space of measurable functions ff on Ω\Omega such that its generalized partial derivatives 1 i 1 n i nf\partial_1^{i_1}\ldots\partial_n^{i_n} f (e.g. in the sense of generalized functions) for all multiindices i=(i 1,,i n) 0 ni = (i_1,\ldots, i_n)\in\mathbb{Z}^n_{\geq 0} with i 1++i nki_1+\ldots +i_n\leq k are in L p(Ω)L_p(\Omega). The most important case is the case of the Sobolev spaces H k(Ω):=W 2 k(Ω)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

Revised on March 7, 2013 19:57:24 by Zoran Škoda (161.53.130.104)