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

Mark A. Peletier; Robert Planqué; Matthias Röger

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 499-510
  • ISSN: 0391-173X

Abstract

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We derive a new criterion for a real-valued function u to be in the Sobolev space W 1 , 2 ( ℝ 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: u ∈ 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.

How to cite

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Peletier, Mark A., Planqué, Robert, and Röger, Matthias. "Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 499-510. <http://eudml.org/doc/272300>.

@article{Peletier2007,
abstract = {We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^\{1,2\}(\mathbb \{R\}^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ 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: $u\in W^\{1,2\}$ 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.},
author = {Peletier, Mark A., Planqué, Robert, Röger, Matthias},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Sobolev regularity; Jensen inequality; rate of convergence; convolution},
language = {eng},
number = {4},
pages = {499-510},
publisher = {Scuola Normale Superiore, Pisa},
title = {Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality},
url = {http://eudml.org/doc/272300},
volume = {6},
year = {2007},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 499
EP - 510
AB - We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\mathbb {R}^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ 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: $u\in W^{1,2}$ 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.
LA - eng
KW - Sobolev regularity; Jensen inequality; rate of convergence; convolution
UR - http://eudml.org/doc/272300
ER -

References

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  1. [1] J. G. Blom and M. A. Peletier, A continuum model of lipid bilayers, European J. Appl. Math.15 (2004), 487–508. Zbl1072.74050MR2115469
  2. [2] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, In: “Optimal Control and Partial Differential Equations” José Luis, Menaldi et. al. (eds.), Amsterdam, IOS Press; Tokyo, Ohmsha, 2001, 439–455. Zbl1103.46310
  3. [3] R. Planqué, “Constraints in Applied Mathematics: Rods, Membranes, and Cuckoos”, Ph.D. thesis, Technische Universiteit Delft, 2005. 
  4. [4] A. C. Ponce, A new approach to Sobolev spaces and connections to Γ -convergence, Calc. Var. Partial Differential Equations19 (2004), 229–255. MR2033060

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