In fracture mechanics one might end up dealing with functions such as $$w(r,\theta) = \sqrt{r} \sin \frac \theta 2,$$ which is defined, for example, on a cracked unit circle, $\Omega = B(0,1)\setminus\{(x,0)\,|\,-1\leq x < 0\}$, and where $(r,\theta)$ is the polar coordinate system originated at $(0,0)$ and oriented "in the standard fashion", i.e. $\theta=0$ corresponds to the positive $x$-axis.

It is quite often remarked in related mathematical literature that $w \in H^{3/2-\epsilon}(\Omega)$, $\forall \epsilon > 0$, and this is something I'd like to show explicitly. Here $H^s(\Omega)$, $s\in \mathbb{R}_+$, denotes the fractional-order Sobolev space $W^{s,2}(\Omega)$.

The unfortunate thing is that the fractional-order Sobolev space is often not defined rigorously in these texts and only a citation to Adams - Sobolev Spaces or to some similar classical text is done. Fortunately, I'm quite certain that the definition is either through the Fourier transform or through the Slobodeckij seminorm (cf. Wikipedia article Sobolev space). The latter approach I've tried with unsatisfactory results. Such an approach involves evaluation of integrals of the form $$\int_\Omega \int_\Omega \frac{|\partial_x w(\boldsymbol{x})-\partial_x w(\boldsymbol{y})|^2}{|\boldsymbol{x}-\boldsymbol{y}|^3} \, \mathrm{d}\boldsymbol{x}\,\mathrm{d}\boldsymbol{y},$$ which, however, turned out to be a challenging task.

Is there any simple approach for showing that $w \in H^{3/2-\epsilon}(\Omega)$?

  • $\begingroup$ I'm a little late here, but there's a really nice proof that the fractional Sobolev norm defined using the Fourier transform is equivalent to the Slobodeckij norm in chapter 25 of Francois Treves's Basic Linear Partial Differential Equations. $\endgroup$ Apr 1, 2019 at 20:42


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