Sobolev regularity via the convergence rate of convolutions and Jensen's inequality
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 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 ( n ). 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: uW 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
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     title = {Sobolev regularity via the convergence rate of convolutions and {Jensen's} inequality},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {499--510},
     publisher = {Scuola Normale Superiore, Pisa},
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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, Serie 5, Volume 6 (2007) no. 4, pp. 499-510. http://archive.numdam.org/item/ASNSP_2007_5_6_4_499_0/

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