Sobolev space

Summary

A Sobolev space $W^{k,p}(U)$ consists of the functions in the Lp space $L^p(U)$ whose weak derivatives up to order $k$ also lie in $L^p(U)$. With the sum of the $L^p$ norms of the derivatives it is a Banach space, and for $p = 2$ it is a Hilbert space, written $H^k(U)$. Sobolev spaces are the main setting of the modern theory of partial differential equations, because weak solutions are found in them by functional analysis. Embedding theorems then translate the integrability of derivatives into integrability or continuity of the function itself.