Sobolev regularity via the convergence rate of convolutions and Jensen's inequality

We derive a new criterion for a real-valued function u to be in the Sobolev space W1,2(Rn). This criterion consists of comparing the value of a functional ¿ f(u) 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 ¿ W1,2 if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the Euler-Lagrange equation.