June  2013, 2(2): 301-318. doi: 10.3934/eect.2013.2.301

Higher differentiability in the context of Besov spaces for a class of nonlocal functionals

1. 

University of Nebraskat-Lincoln, Department of Mathematics, 203 Avery Hall, PO BOX 880130, Lincoln NE 68588-0130, United States, United States

Received  November 2012 Revised  January 2013 Published  March 2013

The aim of this paper is to contribute to the nonlocal theory within the calculus of variations by studying two classes of nonlocal functionals. Since the nonlocal theory is not quite as developed as the local theory, a proof for the existence and uniqueness of minimizers is provided. However, the main result within the paper establishes the higher differentiability, in the context of Besov spaces, for minimizers of nonlocal functionals. This result is obtained under quadratic growth assumptions via the difference quotient method.
Citation: Mikil Foss, Joe Geisbauer. Higher differentiability in the context of Besov spaces for a class of nonlocal functionals. Evolution Equations and Control Theory, 2013, 2 (2) : 301-318. doi: 10.3934/eect.2013.2.301
References:
[1]

Tsegaye G. Ayele and Abraham N. Abebe, Properties of iterated norms in Nikol'skii-Besov type spaces with generalized smoothness, Eurasian Mathematics Journal, 1 (2010), 20-31.

[2]

Viktor I. Burenkov, A theorem on iterated norms for Nikol'skii-Besov spaces and its application, (Russian) Trudy Mat. Inst. Steklov., 181 (1988), 27-39; Translated in Proceedings of the Steklov Institute of Mathematics, (1989), 29-42.

[3]

Viktor I. Burenkov, "Sobolev Spaces on Domains," Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 137, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998.

[4]

Bernard Dacorogna, "Direct Methods in the Calculus of Variations," Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.

[5]

Lawrence C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[6]

Guy Gilboa and Stanley Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Modeling & Simulation, 6 (2007), 595-630. doi: 10.1137/060669358.

[7]

Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592.

[8]

Enrico Giusti, "Direct Methods in the Calculus of Variations," World Scientific Publishing Co. Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.

[9]

Brittney Hinds and Petronela Radu, Dirichlet's principle and wellposedness of solutions for a nonlocal $p$-Laplacian system, Applied Mathematics and Computation, 219 (2012), 1411-1419. doi: 10.1016/j.amc.2012.07.045.

[10]

Giovanni Leoni, "A First Course in Sobolev Spaces," Graduate Studies in Mathematics, 105, American Mathematical Society, Providence, RI, 2009.

show all references

References:
[1]

Tsegaye G. Ayele and Abraham N. Abebe, Properties of iterated norms in Nikol'skii-Besov type spaces with generalized smoothness, Eurasian Mathematics Journal, 1 (2010), 20-31.

[2]

Viktor I. Burenkov, A theorem on iterated norms for Nikol'skii-Besov spaces and its application, (Russian) Trudy Mat. Inst. Steklov., 181 (1988), 27-39; Translated in Proceedings of the Steklov Institute of Mathematics, (1989), 29-42.

[3]

Viktor I. Burenkov, "Sobolev Spaces on Domains," Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 137, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998.

[4]

Bernard Dacorogna, "Direct Methods in the Calculus of Variations," Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.

[5]

Lawrence C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[6]

Guy Gilboa and Stanley Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Modeling & Simulation, 6 (2007), 595-630. doi: 10.1137/060669358.

[7]

Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592.

[8]

Enrico Giusti, "Direct Methods in the Calculus of Variations," World Scientific Publishing Co. Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.

[9]

Brittney Hinds and Petronela Radu, Dirichlet's principle and wellposedness of solutions for a nonlocal $p$-Laplacian system, Applied Mathematics and Computation, 219 (2012), 1411-1419. doi: 10.1016/j.amc.2012.07.045.

[10]

Giovanni Leoni, "A First Course in Sobolev Spaces," Graduate Studies in Mathematics, 105, American Mathematical Society, Providence, RI, 2009.

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