## Research interests

I work in the field of Mathematical Analysis, with a particular interest in Calculus of Variations, Geometric Measure Theory, Optimal Transport and Partial Differential Equations.

My research mainly deals with the following

I have

My Erdös Number is 3.

My research mainly deals with the following

**topics**:- regularity of minimal surfaces and clusters in Riemannian and sub-Riemannian spaces;
- fractional Sobolev and \(BV\) spaces and fractional operators;
- curvature properties of non-\(CD\) spaces linked to heat and entropic flows;
- flows associated to non-smooth velocity fields and (incompressible) Euler equations.

I have

**23****coauthors**: R. Alicandro, L. Ambrosio, K. Bessas, A. Braides, E. Brué, L. De Rosa, M. Calzi, G. E. Comi, G. Crippa, V. Franceschi, F. Giannetti, M. Inversi, P. D. Lamberti, M. Latocca, R. Monti, Q.-H. Nguyen, A. Pinamonti, A. Pratelli, L. Rizzi, G. Saracco, C. Saffirio, M. Solci and D. Spector.My Erdös Number is 3.

## Publications (CvGmt, arXiv, ReserchGate)

**Preprints**

[8]

[8]

*On the monotonicity of non-local perimeter of convex bodies*, with F. Giannetti, submitted (2023) [CvGmt, arXiv]

**[7]**

*Topological singularities arising from fractional-gradient energies*, with R. Alicandro, A. Braides and M. Solci, submitted (2023) [CvGmt, arXiv]

**[6]**

*On the monotonicity of weighted perimeters of convex bodies*, with G. Saracco, submitted (2023) [CvGmt, arXiv]

**[5]**

*Existence and stability of weak solutions of the Vlasov--Poisson system in localized Yudovich spaces*, with G. Crippa, M. Inversi and C. Saffirio, submitted (2023) [CvGmt, arXiv]

**[4]**

*On sets with finite distributional fractional perimeter*, with G. E. Comi, submitted (2023) [CvGmt, arXiv]

**[3]**

*Non-local \(BV\) functions and a denoising model with \(L^1\) fidelity*, with K. Bessas, submitted (2022) [CvGmt, arXiv]

**[2]**

*The Cheeger problem in abstract measure spaces*, with F. Franceschi, A. Pinamonti and G. Saracco, submitted (2022) [CvGmt, arXiv]

**[1]**

*An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces*, with G. Crippa, submitted (2021) [CvGmt, arXiv]

**Accepted**

[1]

[1]

*Lagrangian stability for a system of non-local continuity equations under Osgood condition*, with M. Inversi, accepted (2023), to appear on Contemp. Math. [CvGmt, arXiv]

**Published**

**[22]**

*Full double Hölder regularity of the pressure in bounded domains*, with L. De Rosa and M. Latocca, in Int. Math. Res. Not. IMRN (2023), Paper No. rnad197 [CvGmt, arXiv, eprint]

**[21]**

*Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries*, with L. Rizzi, J. Funct. Anal. 285 (2023), no. 9, Paper No. 110099 [CvGmt, arXiv, eprint]

**[20]**

*Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula*, with G. E. Comi, in Boll. Unione Mat. Ital. (2023) [CvGmt, arXiv, eprint]

**[19]**

*Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces*, with L. De Rosa and M. Inversi, in J. Differ. Equ.

**366**(2023), 833–861. [CvGmt, arXiv, eprint]

**[18]**

*On the Steiner property for planar minimizing clusters. The anisotropic case*, with V. Franceschi and A. Pratelli, in J. Éc. polytech. Math.

10 (2023), 989-1045. [CvGmt, arXiv, eprint]

**[17]**

*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. [CvGmt, arXiv, eprint]

**[16]**

*On the Steiner property for planar minimizing clusters. The isotropic case*, with V. Franceschi and A. Pratelli, in Commun. Contemp. Math. 25 (2023), no. 5, Paper No. 2250040, 29 pp. [CvGmt, arXiv, eprint]

**[15]**

*Failure of the local chain rule for the fractional variation*, with G. E. Comi, in Port. Math. 80 (2023), no. 1-2, 1–25. [CvGmt, arXiv, eprint]

**[14]**

*On double Hölder regularity of the hydrodynamic pressure in bounded domains*, with L. De Rosa and M. Latocca, in Calc. Var. Partial Differential Equations 62 (2023), no. 3, Paper No. 85, 31 pp. [CvGmt, arXiv, eprint]

**[13]**

*On the convex components of a set in \(\mathbb{R}^n\)*, with F. Giannetti, in Forum Math. 35 (2023), no. 1, 187–199. [CvGmt, arXiv, eprint]

**[12]**

*Existence and uniqueness theorems for some semi-linear equations on locally finite graphs*, with A. Pinamonti, in Proc. Amer. Math. Soc. 150 (2022), no. 11, 4757–4770. [CvGmt, arXiv, eprint]

**[11]**

*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. [CvGmt, arXiv, eprint]

**[10]**

*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. [CvGmt, arXiv, eprint]

**[9]**

*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. [CvGmt, arXiv, eprint]

**[8]**

*Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups*, in J. Geom. Anal. 32 (2022), no. 4, 136. [CvGmt, arXiv, eprint]

**[7]**

*Heat and entropy flows in Carnot groups*, with L. Ambrosio, in Rev. Mat. Iberoam. 36 (2020), no. 1, 257–290. [CvGmt, arXiv, eprint]

**[6]**

*Symmetric double bubble in the Grushin plane*, with V. Franceschi, in ESAIM Control Optim. Calc. Var. 25 (2019), Paper No. 77, 37 pp. [CvGmt, arXiv, eprint]

**[5]**

*A maximal function characterization of absolutely continuous measures and Sobolev functions*, with E. Bruè and Q.-H. Nguyen, in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 3, 599–614. [CvGmt, arXiv, eprint]

**[4]**

*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. [CvGmt, arXiv, eprint]

**[3]**

*On the monotonicity of perimeter of convex bodies*, in J. Convex Anal. 25 (2018), no. 1, 93–102 [CvGmt, arXiv, eprint]

**[2]**

*Improved Lipschitz approximation of H-perimeter minimizing boundaries*, 2016, with R. Monti, in J. Math. Pures Appl. (9) 108 (2017), no. 3, 372–398. [CvGmt, arXiv, eprint]

**[1]**

*Sobolev subspaces of nowhere bounded functions*, with P. D. Lamberti, in Real Anal. Exchange 41 (2016), no. 2, 367–375. [CvGmt, arXiv, eprint]

## Academic works

**Ph.D. thesis:**

*A Distributional Approach to Fractional Sobolev Spaces and Fractional Variation*, 2020 [pdf]

Advisor: L. Ambrosio

**M.D. thesis:**

*Intrinsic Lipschitz approximation of \(H\)-perimeter minimizing boundaries*, 2016 [pdf]

Advisor: R. Monti. Co-examiner: L. Ambrosio

**B.D. thesis:**

*Equazioni differenziali ordinarie - Dal Teorema di Cauchy–Lipschitz alle teorie di DiPerna–Lions e Crippa–De Lellis*, 2014 [pdf]

Advisor: R. Monti.

## Talks

**Conference talks**

**[11]**

*A distributional approach to fractional Sobolev and \(BV\) spaces*, NonPUB23 - Nonlocal and Nonlinear Partial Differential Equations at the University of Bologna, Bologna, June 9, 2023 [slides]

**[10]**

*Double Hölder regularity of the hydrodynamic pressure in bounded domains*, Recent Advances in Nonlinear Partial Differential Equations,

Catania, May 26, 2023 [slides]

**[9]**

*Non-local BV functions and a denoising model with \(L^1\)-fidelity*, Shape Optimization, Geometric Inequalities, and Related Topics -

*Two days workshop for young researchers in Naples*, Napoli, January 31, 2023 [slides]

**[8]**

*An abstract approach to the Cheeger problem and applications*, Anisotropic isoperimetric problems & related topics, Rome, September 8, 2022 [slides]

**[7]**

*Bakry-Émery curvature condition and entropic inequalities on metric-measure groups*, AMS-SMF-EMS Joint International Meeting 2022, Special session in Sub-Riemannian Geometry, Grenoble, July 21, 2022 [slides]

**[6]**

*A distributional approach to fractional Sobolev spaces and fractional variation,*100 years Unione Matematica Italiana - 800 years Università di Padova, Padova, May 26, 2022 [slides]

**[5]**

*, XXX Convegno Nazionale di Calcolo delle Variazioni, Levico Terme, February 3, 2020 Levico Terme, February 3, 2020 [slides]*

**A distributional approach to fractional Sobolev spaces and fractional variation: asymptotic analysis****[4]**

*A distributional approach to fractional Sobolev spaces and fractional variation*, International Conference on Elliptic and Parabolic Problems,

*Mini-symposium ``Regularity in Nonlinear Elasticity'*', Gaeta, May 24, 2019 [slides]

**[3]**

*Heat and entropy flows in Carnot groups*, Vector Fields, Surfaces and Perimeters in Singular Geometries, Ferrara, February 28, 2018 [slides]

**[2]**

*Improved Lipschitz approximation on \(H\)-perimeter minimizing boundaries*, XXVII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme, February 6, 2017 [slides]

**[1]**

*On the monotonicity of perimeter of convex bodies*, Workshop on Calculus of Variations and Nonlinear PDEs,

*with a special tribute to Francesco Leonetti on the occasion of his 60th birthday*, Università degli Studi del Sannio, Benevento, November 17, 2016 [slides]

**Seminars**

[15]

[15]

*Sub-Riemannian manifolds are not \(\mathsf{CD}\)*, Bonn, September 19, 2023 [notes]

**[14]**

*A distributional approach to fractional Sobolev and \(BV\) functions I*, Louvain-la-Neuve, April 18, 2023 [slides]

**[13]**

*Bakry-Émery curvature condition and entropic inequalities on metric measure groups,*SISSA Analysis Junior Seminar 2021-2022, Trieste, June 10, 2022 [slides

**]**

**[12]**

*An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces*, Nonlinear Analysis Seminar Series, March 15, 2021 (online) [slides, video]

**[11]**

*Bakry–Émery curvature condition and entropic inequalities on metric-measure groups*, Analysis Seminar, Trento, December 2, 2021 [slides]

**[10]**

*An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces*, Corona Seminars, November 24 2021 (online) [slides]

**[9]**

*On the convex components of a set*, Seminari M.A.P., Pisa, November 18, 2021 [blackboard]

**[8]**

*An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces*, Pisa (SNS), November 17, 2021 (hybrid) [slides]

**[7]**

*A distributional approach to fractional spaces and fractional variation*, Ferrara, March 4, 2021 (online) [slides]

**[6]**

*Bakry-Émery curvature condition and entropic inequalities on metric-measure groups*, Hamburg, December 16, 2020 (online) [slides]

**[5]**

*A distributional approach to fractional Sobolev spaces and fractional variation*, Ph.D. defense, Pisa, February 13, 2020 [slides]

**[4]**

*A distributional approach to fractional Sobolev spaces and fractional variation: asymptotic analysis*, Padova, December 11, 2019 [slides]

**[3]**

*A distributional approach to fractional Sobolev spaces and fractional variation*, Basel, April 24 2019 [slides]

**[2]**

*Carnot groups from an RCD point of view: heat, entropy and their properties*, Seminario Dottorandi SNS, Pisa, January 21, 2019 [slides]

**[1]**

*Archimedes, a dinner and a theorem - A divertissement on the monotonicity of perimeter*, Seminari M.A.P., Pisa, April 24, 2018 [blackboard]

## Conferences, meetings and workshops

**[36]**NonPUB23 - Nonlocal and Nonlinear Partial Differential Equations at the University of Bologna, Bologna (Italy), June 8 - 9, 2023

**[35]**Recent Advances in Nonlinear Partial Differential Equations, Catania (Italy), May 24 - 26, 2023

**[34]**XXXII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), May 8 - 12, 2023

**[33]**Geometry and Control in Florence,

*in honor of Andrei Agrachev's 70th birthday*, Cortona (Italy), March 27 - 31, 2023

**[32]**Shape Optimization, Geometric Inequalities, and Related Topics,

*Two days workshop for young researchers in Naples*, Napoli (Italy), January 30 - 31, 2023

**[31]**Workshop on Singular Perturbations and Geometric Structures, Trieste (Italy), November 21 - 23, 2022

**[30]**Anisotropic isoperimetric problems & related topics, Rome (Italy), September 5 - 9, 2022

**[29]**AMS-SMF-EMS Joint International Meeting 2022,

*Special session in Sub-Riemannian Geometry*, Grenoble (France), July 20 - 22, 2022 (attended remotely during quarantine)

**[28]**Geometric Analysis in Metric Measure Spaces,

*Two days in honor of Bruno Franchi and Raul Serapioni*, Levico Terme, June 30 - July 1, 2022

**[27]**12th School on Analysis and Geometry in Metric Spaces, Levico Terme (Italy), June 27 - 29, 2022

**[26]**Isoperimetric problems, Pisa (Italy), 20 - 24 June 2022

**[25]**100 years Unione Matematica Italiana - 800 years Università di Padova, Padova (Italy), May 23 - 27, 2022

**[24]**XXXI Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), May 9 - 13, 2022

**[23]**

*Transport, Fluids and Mixing, Pisa (Italy), January 24 - 28, 2022 (online)*

**[22]**

*Recent Progress in Geometric Analysis and PDEs for the 70th birthday of Bruno Franchi , Bologna (Italy), September 16 - 18 , 2021*

**[21]**PaPa - Padua Paris Sub-Riemannian seminar, Padova (Italy), September 6 - 7, 2021

**[20]**XXX Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), February 3 - 7, 2020

**[19]**Shape Optimization and Isoperimetric and Functional Inequalities, Levico Terme (Italy), September 23 - 27, 2019

**[18]**International Conference on Elliptic and Parabolic Problems, Gaeta (Italy), May 20 - 24, 2019

**[17]**Some topics of Geometric Analysis and Geometric Measure Theory, Pisa (Italy), April 16 -17, 2019

**[16]**Optimal Transport and Geometric Analysis, Venezia (Italy), April 1 - 5, 2019

**[15]**XXIX Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), February 4 - 8, 2019

**[14]**Optimal Transportation and Applications, Pisa (Italy), November 12 - 16, 2018

**[13]**A Sub-Riemannian Day in Padova, Padova (Italy), September 14, 2018

**[12]**Vector Fields, Surfaces and Perimeters in Singular Geometries, Ferrara (Italy), February 27 - 28, 2018

**[11]**XXVIII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), February 12 - 16, 2018

**[10]**Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications, Bologna (Italy), January 24 - 27, 2018

**[9]**Recent Advances in PDEs and the Calculus of Variations, Venezia (Italy), July 3 - 6, 2017

**[8]**CIME-CIRM Course on New Trends on Analysis and Geometry in Metric Spaces (10th School), Levico Terme (Italy), June 26 - 30, 2017

**[7]**XXVII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), February 6 - 10, 2017

**[6]**Workshop on Calculus of Variations and Nonlinear PDEs

*with a special tribute to Francesco Leonetti on the occasion of his 60th birthday,*Università degli Studi del Sannio, Benevento (Italy), November 16 - 18, 2016

**[5]**Optimal Transport and Applications, Scuola Normale Superiore, Pisa (Italy), November 7 - 11, 2016

**[4]**Mini-courses in Mathematical Analysis, Padova (Italy), June 27 - July 1, 2016

**[3]**Singular Phenomena and Singular Geometries, Pisa (Italy), June 20 - 23, 2016

**[2]**Two-day Meeting on linear and nonlinear PDE's

*in honor of the 65th birthday of Cristian Gutierrez*, Accademia delle Scienze, Bologna (Italy), June 9 - 10, 2016

**[1]**XXVI Convegno Nazionale di Calcolo delle Variazioni, Levico Terme (Italy), January 18 - 22, 2016