If $\Omega\subset\mathbf{R}^d$ has a smooth boundary it is known that the distance function $\mathrm{d}_\Omega:x\mapsto \mathrm{d}(x,\partial\Omega)$ is smooth on a neighborhood of $\partial\Omega$. My question is about the hessian of $\mathrm{d}_\Omega$ outside that neighborhood : does it exists in a weaker sense ? I am thinking of Sobolev regularity, for instance. Since $\mathrm{d}_\Omega$ is lipschitz, we know that it belongs to $\mathrm{W}_{\mathrm{loc}}^{1,\infty}(\mathbf{R}^d)$ and my hope is that those distance functions enjoy more regularity (in the whole space) than mere $\mathrm{W}^{1,\infty}_{\mathrm{loc}}(\mathbf{R}^d)$.