We derive a new criterion for a real-valued function to be in the Sobolev space . This criterion consists of comparing the value of a functional with the values of the same functional applied to convolutions of with a Dirac sequence. The difference of these values converges to zero as the convolutions approach , and we prove that the rate of convergence to zero is connected to regularity: 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.
@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}, pages = {499--510}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {4}, year = {2007}, mrnumber = {2394408}, zbl = {1185.46026}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2007_5_6_4_499_0/} }
TY - JOUR AU - Peletier, Mark A. AU - Planqué, Robert AU - Röger, Matthias TI - Sobolev regularity via the convergence rate of convolutions and Jensen's inequality JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 499 EP - 510 VL - 6 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2007_5_6_4_499_0/ LA - en ID - ASNSP_2007_5_6_4_499_0 ER -
%0 Journal Article %A Peletier, Mark A. %A Planqué, Robert %A Röger, Matthias %T Sobolev regularity via the convergence rate of convolutions and Jensen's inequality %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 499-510 %V 6 %N 4 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2007_5_6_4_499_0/ %G en %F 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, 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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