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On a new class of fractional partial differential equations II

Tien-Tsan Shieh and Daniel E. Spector EMAIL logo

Abstract

In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular, we here establish an L1 Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler–Lagrange equations obtained as conditions of minimality. In addition, we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.


Dedicated to Irene Fonseca, on the occasion of her 60th birthday, with esteem and affection



Communicated by Juha Kinnunen


Funding statement: The first author is partially supported by National Science Council of Taiwan under research grant NSC 101-2115-M-009-014-MY3. The second author is supported by the Taiwan Ministry of Science and Technology under research grants 103-2115-M-009-016-MY2 and 105-2115-M-009-004-MY2.

Acknowledgements

The first author would like to thank Professor Ming-Chih Lai for his support during a portion of this project. The second author would like to thank Eliot Fried for discussions regarding non-local continuum mechanics and also for making him aware of the work of Edelen and Laws, Edelen, Green and Laws. The authors would like to thank Armin Schikorra for many helpful discussions regarding the paper. Finally, the authors would like to thank the referees for their comments, which have improved the clarity of presentation in the paper.

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Received: 2016-11-22
Revised: 2017-03-14
Accepted: 2017-03-17
Published Online: 2017-04-19
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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