Sobolev regularity via the convergence rate of convolutions and Jensen's inequality
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 4, p. 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.
Classification:  46E35,  49J45,  49J40
@article{ASNSP_2007_5_6_4_499_0,
author = {Peletier, Mark A. and Planqu\'e, Robert and R\"oger, Matthias},
title = {Sobolev regularity via the convergence rate of convolutions and Jensen's inequality},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {4},
year = {2007},
pages = {499-510},
zbl = {1185.46026},
mrnumber = {2394408},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_4_499_0}
}

Peletier, Mark A.; Planqué, Robert; Röger, Matthias. Sobolev regularity via the convergence rate of convolutions and Jensen's inequality. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 4, pp. 499-510. http://www.numdam.org/item/ASNSP_2007_5_6_4_499_0/

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