We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}\left({ℝ}^n\right)$. This criterion consists of comparing the value of a functional $\int f\left(u\right)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization...

Let $M$ be a Riemannian manifold, which possesses a transitive Lie group $G$ of isometries. We suppose that $G$, and therefore $M$, are compact and connected. We characterize the Sobolev spaces $W_p^1\left(M\right)$