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Sobolev spaces are function spaces generalizing the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, functions in a Sobolev space have also differentiability conditions: that is, we require all partial derivatives of the function up to a certain order belong to a certain fixed Lebesgue space.

More particularly, if $U \subseteq \mathbb{R}^n$ is an open set, the Sobolev space $W^{k, p}(U)$ consists of those functions $f$ such that for every multiindex $\alpha$ of length at most $k$, every weak partial derivative $D^{\alpha}f$ exists and is an element of $L^p$.

The Sobolev spaces are equipped with norms defined by

$$\|u\| = \left( \sum_{|\alpha| \le k} \|D^{\alpha} u\|_{L^p(U)}^p \right)^{1/p}$$

if $p < \infty$, and

$$\|u\| = \max_{|\alpha| \le k} \|D^{\alpha}\|_{L^{\infty}(U)}$$

if $p = \infty$. These norms measure both the size and the regularity of a function.

Reference: Sobolev space.

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