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Advances in Calculus of Variations

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Volume 11, Issue 3


On a new class of fractional partial differential equations II

Tien-Tsan Shieh / Daniel E. Spector
  • Corresponding author
  • Department of Applied Mathematics, National Chiao Tung University, Hsinchu,Taiwan; and National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei,106, Taiwan
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Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/acv-2016-0056


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.

Keywords: Fractional gradient; fractional Hardy inequality; fractional partial differential equations; interpolation; Dirichlet forms

MSC 2010: 26A33; 35R11; 49J45; 35JXX

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


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About the article

Received: 2016-11-22

Revised: 2017-03-14

Accepted: 2017-03-17

Published Online: 2017-04-19

Published in Print: 2018-07-01

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.

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 3, Pages 289–307, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2016-0056.

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