# Weak Differentiability of Distance Function

Let $\Omega$ be a bounded uniformly convex domain in $R^n$ with smooth ($C^\infty$ boundary). Let $\rho(x) = d(x,\partial\Omega)$ for $x\in \Omega$. Gilbarg and Trudinger 14.6 gives us smoothness on a domain near the boundary. $\rho$ is also Lipschitz everywhere, and hence by Rademacher, differentiable almost everywhere. I also know that since the domain is uniformly convex, $\rho$ is concave.

What I want is $\rho$ to be in $W^{1,2}_0(\Omega)$ or $H^1_0(\Omega)$.

I also want to be able to write

$\nabla \log\rho(y) - \nabla\log \rho(x) = \int_{0}^1 \nabla^2 \log\rho|_{ty+(1-t)x}(y-x,y-x) dt.$

I know that the above relation is clearly satisfied for smooth functions (gradient field theorem), but is it satisfied for weaker conditions?

For reference - I am trying to understand Appendix B in http://arxiv.org/pdf/1407.0526v1.pdf.

Thanks.

• Lipschitz property implies $W^{1,p}$ for all $p<\infty$. Logarithm preserves Sobolev class membership as long as the domain stays away from the zero set of the function.
– user147263
Sep 18 '15 at 5:30