LisMath Seminar 

06/05/2020, 17:00 — 18:00 — Online
Catharine Lo, LisMath, Faculdade de Ciências, Universidade de Lisboa

On a new class of fractional partial differential equations

In this talk, I will discuss a class of fractional partial differential equations. Such fractional partial differential equations are obtained from extending the theory regarding the Riesz fractional gradients. I will first introduce the fractional differential operators $\nabla^s$ and $\div^s$. I will then explain 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. These ideas can then be used to establish analogous results for fractional partial differential equations.

Bibilography:

[1] Giovanni E. Comi and G. Stefani. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I. 2019. arXiv: 1910.13419 [math.FA].

[2] Giovanni E. Comi and Giorgio Stefani. A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up. In: Journal of Functional Analysis 277.10 (2019), pp. 3373- 3435. issn: 0022-1236. doi: 10.1016/j.jfa.2019.03.011.

[3] José Francisco Rodrigues and Lisa Santos. On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient. In: Applied Mathematics & Optimization 80, no. 3 (2019), pp. 835?852. doi: 10.1007/s00245- 019-09610-0.

[4] Tien-Tsan Shieh and Daniel Spector. On a new class of fractional partial differential equations. In: Advances in Calculus of Variations 8 (2015), pp. 321-366. doi: 10.1515/acv-2014-0009.

[5] Tien-Tsan Shieh and Daniel Spector. On a new class of fractional partial differential equations II. In: Advances in Calculus of Variations 11 (2018), pp. 289-307. doi: 10.1515/acv-2016-0056.

[6] Miroslav Silhavy. Fractional vector analysis based on invariance require- ments (critique of coordinate approaches). In: Continuum Mechanics and Thermodynamics 32, Issue 1 (2020), pp. 207-288. doi: 10.1007/s00161-019- 00797-9.

See also

LisMath_seminar_presentation.pdf