On a new class of fractional partial differential equations II

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 Hardy inequality, obtain further regularity results for s...

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Published inAdvances in calculus of variations Vol. 11; no. 3; pp. 289 - 307
Main Authors Shieh, Tien-Tsan, Spector, Daniel E.
Format Journal Article
LanguageEnglish
Published Berlin De Gruyter 01.07.2018
Walter de Gruyter GmbH
Subjects
26A33
35JXX
35R11
49J45
Calculus of variations
Dependence
Differential calculus
Dirichlet forms
Dirichlet problem
Euler-Lagrange equation
Fractional gradient
fractional Hardy inequality
fractional partial differential equations
Functionals
Interpolation
Mathematical analysis
Partial differential equations
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Summary: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 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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:1864-8258
1864-8266
DOI:10.1515/acv-2016-0056