STARS Starting Grant 2023 New Directions in Fractional Calculus - NewFrac
The research project focuses on Fractional Calculus, a branch of Mathematical Analysis that extends traditional calculus to operators of non-integer order and has broad applications across diverse fields. The project centers on a novel concept in this field: a distributional theory of fractional spaces, which has recently gained attention for its potential to solve complex problems in Geometric Measure Theory, Functional Analysis, and PDE Theory. We aim to build upon this new distributional approach by exploring three key objectives:
The project NewFrac was awarded the STARS Seal of Excellence in the Starting Grant category, along with € 225,000 in funding for a duration of 2.5 years, by the University of Padova (Italy).
- Investigating the geometric properties of sets with finite distributional fractional perimeter.
- Advancing the theory of distributional fractional spaces, especially the one of functions with bounded distributional variation.
- Applying this distributional framework to solve fractional PDEs and study the properties of the solutions.
The project NewFrac was awarded the STARS Seal of Excellence in the Starting Grant category, along with € 225,000 in funding for a duration of 2.5 years, by the University of Padova (Italy).
Team members
Publications
The list below provides the complete set of publications by the PI and collaborators related to this theory, as well as some closely related works by the PI and collaborators in non-local analysis.
Publications directly related to project
[8] On sets with finite distributional fractional perimeter, with G. E. Comi, In: Anisotropic Isoperimetric Problems and Related Topics, V. Franceschi, A. Pluda e G. Saracco (eds.), INdAM 2022, Springer INdAM Series, vol. 62, Springer, Singapore [arXiv, DOI]
[7] Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula, with G. E. Comi, in Boll. Unione Mat. Ital. 17 (2024), 259-281 [arXiv, DOI]
[6] A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I, with G. E. Comi, in Rev. Mat. Complut. 36 (2023), no. 2, 491–569. [arXiv, DOI]
[5] Failure of the local chain rule for the fractional variation, with G. E. Comi, in Port. Math. 80 (2023), no. 1-2, 1–25. [arXiv, DOI]
[4] A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II, with E. Bruè, M. Calzi and G. E. Comi, in C. R. Math. Acad. Sci. Paris 360 (2022), 589–626. [arXiv, DOI]
[3] The fractional variation and the precise representative of \(BV^{\alpha,p}\) functions, with G. E. Comi and D. Spector, in Fract. Calc. Appl. Anal. 25 (2022), no. 2, 520–558. [arXiv, DOI]
[2] Leibniz rules and Gauss-Green formulas in distributional fractional spaces, with G. E. Comi, in J. Math. Anal. Appl. 514 (2022), no. 2, Paper No. 126312, 41 pp. [arXiv, DOI]
[1] A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up, with G. E. Comi, in J. Funct. Anal. 277 (2019), no. 10, 3373–3435. [arXiv, DOI]
Other publications closely related to the project
[7] Sharp conditions for the BBM formula and asymptotics of heat content-type energies, with L. Gennaioli, submitted (2025) [CvGmt]
[6] A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodeckiĭ spaces, with F. Bianchi and A. C. Zagati, submitted (2024) [arXiv]
[5] Topological singularities arising from fractional-gradient energies, with R. Alicandro, A. Braides and M. Solci, submitted (2023) [arXiv]
[4] On the monotonicity of non-local perimeter of convex bodies, with F. Giannetti, Topol. Methods Nonlinear Anal. 64 (2024), no. 2, 693–710 [arXiv, DOI]
[3] On the \(N\)-Cheeger problem for component-wise increasing norms, with G. Saracco, J. Math. Pures Appl. (9) 189 (2024), 103593 [arXiv, DOI]
[2] Non-local \(BV\) functions and a denoising model with \(L^1\) fidelity, with K. Bessas, Adv. Calc. Var. 18 (2025), no. 1, 189–217
[arXiv, DOI]
[1] The Cheeger problem in abstract measure spaces, with F. Franceschi, A. Pinamonti and G. Saracco, in J. London Math. Soc. 109 (2024), no. 1, Paper No. e12840, 55 pp. [arXiv, DOI]
Publications directly related to project
[8] On sets with finite distributional fractional perimeter, with G. E. Comi, In: Anisotropic Isoperimetric Problems and Related Topics, V. Franceschi, A. Pluda e G. Saracco (eds.), INdAM 2022, Springer INdAM Series, vol. 62, Springer, Singapore [arXiv, DOI]
[7] Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula, with G. E. Comi, in Boll. Unione Mat. Ital. 17 (2024), 259-281 [arXiv, DOI]
[6] A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I, with G. E. Comi, in Rev. Mat. Complut. 36 (2023), no. 2, 491–569. [arXiv, DOI]
[5] Failure of the local chain rule for the fractional variation, with G. E. Comi, in Port. Math. 80 (2023), no. 1-2, 1–25. [arXiv, DOI]
[4] A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II, with E. Bruè, M. Calzi and G. E. Comi, in C. R. Math. Acad. Sci. Paris 360 (2022), 589–626. [arXiv, DOI]
[3] The fractional variation and the precise representative of \(BV^{\alpha,p}\) functions, with G. E. Comi and D. Spector, in Fract. Calc. Appl. Anal. 25 (2022), no. 2, 520–558. [arXiv, DOI]
[2] Leibniz rules and Gauss-Green formulas in distributional fractional spaces, with G. E. Comi, in J. Math. Anal. Appl. 514 (2022), no. 2, Paper No. 126312, 41 pp. [arXiv, DOI]
[1] A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up, with G. E. Comi, in J. Funct. Anal. 277 (2019), no. 10, 3373–3435. [arXiv, DOI]
Other publications closely related to the project
[7] Sharp conditions for the BBM formula and asymptotics of heat content-type energies, with L. Gennaioli, submitted (2025) [CvGmt]
[6] A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodeckiĭ spaces, with F. Bianchi and A. C. Zagati, submitted (2024) [arXiv]
[5] Topological singularities arising from fractional-gradient energies, with R. Alicandro, A. Braides and M. Solci, submitted (2023) [arXiv]
[4] On the monotonicity of non-local perimeter of convex bodies, with F. Giannetti, Topol. Methods Nonlinear Anal. 64 (2024), no. 2, 693–710 [arXiv, DOI]
[3] On the \(N\)-Cheeger problem for component-wise increasing norms, with G. Saracco, J. Math. Pures Appl. (9) 189 (2024), 103593 [arXiv, DOI]
[2] Non-local \(BV\) functions and a denoising model with \(L^1\) fidelity, with K. Bessas, Adv. Calc. Var. 18 (2025), no. 1, 189–217
[arXiv, DOI]
[1] The Cheeger problem in abstract measure spaces, with F. Franceschi, A. Pinamonti and G. Saracco, in J. London Math. Soc. 109 (2024), no. 1, Paper No. e12840, 55 pp. [arXiv, DOI]
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