Paper Info
Title
Can the Nonlocal Characterization of Sobolev Spaces by Bourgain et al. Be Useful for Solving Variational Problems?
Abstract
We question whether the recent characterization of Sobolev spaces by Bourgain, Brezis, and Mironescu (2001) could be useful to solve variational problems on $W^{1,p}(\Omega)$. To answer this, we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier. Then, for the approximated formulation, we prove existence, uniqueness, and convergence of the solution to the unique solution of the initial formulation. We show that these results can also be extended in the $BV$-case. Interestingly, this approximation leads to a unified implementation, for Sobolev spaces (including with high $p$-values) and for the $BV$ space. Finally, we show how this theoretical study can indeed lead to a numerically tractable implementation, and we give some image diffusion results as an illustration.
Year
DOI
Venue
2009
10.1137/070696751
SIAM J. Numerical Analysis
Keywords
Field
DocType
unique solution,differential quotient,nonlocal characterization,radial mollifier,image diffusion result,approximated formulation,tractable implementation,sobolev space,sobolev spaces,variational problems,initial formulation,integral operator,unified implementation,calculus of variation,functional analysis
Uniqueness,Mathematical optimization,Interpolation space,Mathematical analysis,Calculus of variations,Quotient,Sobolev space,Sobolev inequality,Mollifier,Sobolev spaces for planar domains,Mathematics
Journal
Volume
Issue
ISSN
47
2
0036-1429
Citations 
PageRank 
References 
10
2.18
6
Authors
2
Name
Order
Citations
PageRank
Gilles Aubert11275108.17
Pierre Kornprobst239936.67