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

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