On a new class of fractional partial differential equations

By Tien-Tsan Shieh, Daniel Spector. 2014

137 citations recorded by the source corpus.

Abstract In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

Subfields: Applied Mathematics, Mechanics of Materials, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/3fde1d1cd2709653d8cd2722bf37ec75fdecf3c4

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In Applied Mathematics

In Mechanics of Materials

In Modeling and Simulation

Citations in the corpus

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up (2018, methodology): “In this paper, we want to combine the functional approach of [17, 18] with the distributional point of view of [19] to develop a satisfactory theory extending De Giorgi’s approach to variation and perimeter in the fractional setting.”
An $L^1$-type estimate for Riesz potentials (2014, background): “Theorem A is false when N = 1 , see Counterexample 3.2 in Section 3. Our motivation for such an inequality can be found in the study of certain fractional partial differential equations introduced in [20], where existence results are demonstrated for a continuous spectrum of such equations parameter”
Fractional Piola identity and polyconvexity in fractional spaces (2018, methodology): “15] also pays attention to the s-fractional gradient, providing a proof not based on Fourier transform of a fractional fundamental theorem of calculus, also proved in [32], which is used for showing a Sobolev type inequality from Hs,p to Lp.”