Computational Mechanics

A function defined on the domain $\Omega$ is said to have Sobolev regularity $W^{r, 2}:=H^r$ if all its derivatives up to the order $r$ belong to $L^2(\Omega)$. Specifically, this implies that

$\int_{\Omega}\left|\frac{\partial^k}{\partial x^k} f(x)\right|^2<\infty, \text { for all } k \leq r$

The largest $r$ for which this condition is met is the Sobolev regularity of the function.

This concept can be extended to $d$-variate functions by considering the partial derivatives $\partial \alpha f$, where $\alpha=\left(\alpha_1, \ldots, \alpha_d\right)$ is a tuple, and it satisfies

$\sum_k \alpha_k \leq r$

When we compute $\frac{\partial^k}{\partial x^k} f(x)$, we may have to deal with non-integer derivatives, there's a notion called "fractional Sobolev regularity". These derivatives can be defined using the Fourier transform.

In simple terms, the Sobolev regularity essentially describes a certain "smoothness" of a function. This concept applies not only to univariate functions but also to multivariate ones. The fractional Sobolev regularity is an extension of this idea, accounting for non-integer order derivatives.