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

Peletier, Mark A.; Planqué, Robert; Röger, Matthias

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 499-510.

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 problem with constraints, where regularity of minimizers cannot be deduced from the Euler-Lagrange equation.

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