Welcome to the Home Page of the Research Group in Calculus of Variations and Geometric Measure Theory at Pisa. Look here to know more about us. Please, feel free to download and use all the material you find interesting. If you want to receive regularly the group news, subscribe to our mailing list in the mail section.

If you want to receive an account to insert your preprints, fill-in the account request.

- Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture (Gianni Dal Maso)
- Anisotropic geometric variational problems: existence, regularity and uniqueness (Antonio De Rosa)
- Analytic Techniques for Geometric and Functional Inequalities (Nicola Fusco)
- Geometry of Metric Groups (Enrico Le Donne) > papers
- Everything You Always Wanted to Know About the JKO Scheme (Filippo Santambrogio)
- Variational approach to the regularity of the free boundaries (Bozhidar Velichkov)

- 4 post-doc positions at the Scuola Superiore Meridionale (deadline: 15 Oct 2024)
- PhD Position at Goethe-University Frankfurt (deadline: 31 Oct 2024)
- Doctoral and postdoctoral positions at Mathematics Münster (deadline: 1 Nov 2024)
- ERC Postdoc position in sub-Riemannian geometry at SISSA, Trieste (Italy) (deadline: 6 Nov 2024)
- PhD position in differential geometry Frankfurt (deadline: 15 Nov 2024)
- 6 year research position in Lisbon (deadline: 29 Nov 2024)
- 2 ERC Postdoc positions at Bocconi University, Milan (deadline: 30 Nov 2024)
- Assistant Professor at Bocconi University (deadline: 15 Dec 2024)
- Expression of interest for a two-year postdoctoral position in Padova (Italy) (deadline: 31 Dec 2024)

- 23 Oct 2024:
**M. J. Gursky:***Some rigidity results for asymptotically hyperbolic Einstein metrics*

- 14 - 18 Oct 2024: Mathematical Materials Science: Defects and Polycrystals ICMS, Edinburgh
- 23 - 25 Oct 2024: Nonlinear partial differential equations in Salzburg Salzburg, Austria
- 28 - 29 Oct 2024: PDEs in Milan 2024 University of Milan
- 28 - 31 Oct 2024: Fall School "Boundary Analysis for Dispersive and Viscous Fluids" GSSI, L'Aquila, Italy
- 4 - 7 Nov 2024: conference "Spectral Theory and Differential Equations" University of Hradec Králové, Czech Republic
- 5 Nov - 5 Dec 2024: Γ-CONVERGENCE: A 50 YEARS LONG STORY Universitat de Barcelona
- 13 - 15 Nov 2024: PDE Festival@UniMi Università degli Studi di Milano, Room C11, Via Mangiagalli 25, Milano (Italy)
- 20 - 21 Nov 2024: Calculus of Variations and Free Boundary Problems IX Aula Magna, Dipartimento di Matematica, Università di Pisa
- 25 - 29 Nov 2024: Frontiers in Sub-Riemannian Geometry (Aux frontières de la géométrie sous-riemannienne) CIRM, Marseille (France)
- 26 - 28 Nov 2024: (Non)local (non)linear workshop I in Prague Prague, Czech Republic
- 28 - 29 Nov 2024: Workshop on Geometric properties and regularity in variational problems Pavia
- 2 - 6 Dec 2024: Optimal Transportation and Applications Centro di Ricerca Matematica Ennio De Giorgi, Pisa
- 18 - 20 Dec 2024: ASK Conference 2024 Università di Bologna, Italy
- 19 - 25 Jan 2025: Taiwan Winter School in Analysis Taipei
- 20 Jan - 7 Feb 2025: Calculus of Variations and PDE's in Geometric Analysis Sapienza, Università di Roma
- 10 - 14 Feb 2025: XXXIV Convegno Nazionale di Calcolo delle Variazioni Hotel Mediterraneo, Riccione
- 31 Mar - 4 Jul 2025: Intensive trimester at GSSI on "Particles, Fluids and Patterns: Analytical and Computational Challenges" Gran Sasso Science Institute, L'Aquila, Italy
- 1 - 7 Jun 2025: Spring School on Analysis 2025: Function Spaces and Applications XIII Paseky nad Jizerou, Czech Republic
- 16 - 20 Jun 2025: Vito Volterra Meeting in Calculus of Variations 2025
- 23 - 27 Jun 2025: Topics in Geometric Analysis Università di Pisa
- 23 - 27 Jun 2025: Workshop: Novel methods of PDEs and function spaces University of Warsaw
- 30 Jun - 4 Jul 2025: Topics in Geometric Measure Theory Università di Pisa
- 13 - 17 Jul 2025: Warsaw Summer School on Evolutionary PDEs University of Warsaw
- 9 - 13 Sep 2025: Diffusion in Warsaw 2025

- S. Borza - M. Magnabosco - T. Rossi - K. Tashiro (preprint)

The curvature exponent of sub-Finsler Heisenberg groups (2024) - G. Bevilacqua - L. Lussardi - A. Marzocchi (preprint)

Soap films: from the Plateau problem to deformable boundaries (2024) - S. Eberle - A. Figalli - G. S. Weiss (Accepted Paper:
*Ann. of Math.*)

Complete classification of global solutions to the obstacle problem (2024) - I. Fleschler - R. Resende (Preprint)

On the regularity of area minimizing currents at boundaries with arbitrary multiplicity (2024) - I. Fleschler (Preprint)

On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity (2024) - S. Honda - A. Mondino (Submitted Paper)

Gap phenomena under curvature restrictions (2024) - N. Gigli - S. Vincini (preprint)

Stability of the heat flow under convergence in concentration and consequences (2024) - M. Di Marco - A. Pinamonti - D. Vittone - K. Zambanini (Preprint)

Stepanov Differentiability Theorem for intrinsic graphs in Heisenberg groups (2024) - A. Figalli - J. P. G. Ramos (Submitted Paper)

Improved stability versions of the Prékopa–Leindler inequality (2024) - T. Schmidt - J. H. Schütt (Submitted Paper)

Partial regularity for variational integrals with Morrey-Hölder zero-order terms, and the limit exponent in Massari's regularity theorem (2024) - M. Cicalese - T. Heilmann (Preprint)

Surfactants in a non-local model for phase transitions (2024) - A. Guerra - X. Lamy - K. Zemas (Preprint)

On the existence of degenerate solutions of the two-dimensional H-system (2024) - G. Giovannardi - A. Pinamonti - S. Verzellesi (Preprint)

Curvature estimates for minimal hypersurfaces in the Heisenberg group (2024) - C. Alessi - L. Brasco - M. Miranda Jr (Preprint)

A Schroedinger operator with confining potential having quadratic growth (2024) - B. Han (Preprint)

ABP estimate on metric measure spaces via optimal transport (2024) - B. Han (Preprint)

Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature (2024) - B. Han (
*Journal of Functional Analysis*)

On the asymptotic behaviour of the fractional Sobolev seminorms: a geometric approach (2024) - B. Han - Z. Xu (
*Calculus of Variations and Partial Differential Equations*)

Sharp uncertainty principles on metric measure spaces (2024) - B. Han - D. Liu - Z. Zhu (Preprint)

On the geometry of Wasserstein barycenter I (2024) - N. Guglielmi - F. P. Maiale (Submitted Paper)

Gripenberg-like algorithm for the lower spectral radius (2024)

- M. Novaga - E. Paolini - V. M. Tortorelli (Accepted Paper:
*Trans. Amer. Math. Soc.*)

Locally isoperimetric partitions (2023) - G. Bellettini - R. Marziani - R. Scala (
*Journal de l’École polytechnique — Mathématiques*)

A non-parametric Plateau problem with partial free boundary (2024) - A. Braides - G. Dal Maso - C. Le Bris (Accepted Paper:
*Ann. Inst. H. Poincaré Anal. Non Linéaire*)

A closure theorem for $\Gamma$-convergence and H-convergence with applications to non-periodic homogenization (2024) - E. G. Tolotti (preprint)

Stability of the Von Kármán regime for thin plates under Neumann boundary conditions (2024) - J. Candau-Tilh - M. Goldman - B. Merlet (Accepted Paper:
*Calc. Var. and PDEs*)

An exterior optimal transport problem (2024) - P. Dondl - A. Maione - S. Wolff-Vorbeck (
*ESAIM: Control, Optimisation and Calculus of Variations*)

Phase field model for multi-material shape optimization of inextensible rods (2024) - M. Santilli (
*IMRN*)

Finite total curvature and soap bubbles with almost constant higher-order mean curvature (2024) - G. De Philippis - A. R. Mészáros - F. Santambrogio - B. Velichkov (
*Arch. Rat. Mech. Anal.*)

BV Estimates in Optimal Transportation and Applications (2015) - D. Mazzoleni - G. Tortone - B. Velichkov (
*J. Convex Analysis: volume dedicated to Giuseppe Buttazzo on the occasion of his 70th birthday.*)

On the dimension of the singular set in optimization problems with measure constraint (2024) - L. Ferreri - B. Velichkov (Accepted Paper:
*Trans. Amer. Math. Soc.*)

Regularity for one-phase Bernoulli problems with discontinuous weights and applications (2023) - G. Buttazzo - M. van den Berg - B. Velichkov (Proceedings:
*Chapter in the book "New Trends in Shape Optimization"*)

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity (2015) - M. Colombo - L. Spolaor - B. Velichkov (
*Geom. Funct. Anal.*)

A logarithmic epiperimetric inequality for the obstacle problem (2018) - G. Buttazzo - B. Velichkov (
*SIAM J. Math. Anal.*)

A Shape Optimal Control Problem with Changing Sign Data (2018) - D. Bucur - I. Fragalà - G. Verzini - B. Velichkov (
*Trans. Amer. Math. Soc.*)

On the honeycomb conjecture for a class of minimal convex partitions (2018) - D. Mazzoleni - S. Terracini - B. Velichkov (
*Analysis & PDE*)

Regularity of the free boundary for the vectorial Bernoulli problem (2020) - L. Spolaor - B. Velichkov (
*Comm. Pure Appl. Math.*)

An epiperimetric inequality for the regularity of some free boundary problems: the $2$-dimensional case (2019) - M. Colombo - L. Spolaor - B. Velichkov (
*J. Reine Angew. Math.*)

On the asymptotic behavior of the solutions to parabolic variational inequalities (2020) - M. Engelstein - L. Spolaor - B. Velichkov (
*Duke Math. J.*)

Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional (2020) - M. Colombo - L. Spolaor - B. Velichkov (
*Comm. Pure Appl. Math.*)

Direct epiperimetric inequalities for the thin obstacle problem and applications (2020) - L. Spolaor - B. Trey - B. Velichkov (
*Communications in Partial Differential Equations*)

Free boundary regularity for a multiphase shape optimization problem (2018) - M. Engelstein - L. Spolaor - B. Velichkov (
*Geom. Topol.*)

(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents (2019) - F. Générau - E. Oudet - B. Velichkov (
*ESAIM: M2AN*)

Numerical computation of the cut locus via a variational approximation of the distance function (2022) - F. P. Maiale - G. Tortone - B. Velichkov (
*Annali della Scuola Normale Superiore di Pisa, Classe di Scienze*)

The boundary Harnack principle on optimal domains (2024) - B. Velichkov (Lecture Notes:
*Lecture Notes of the Unione Matematica Italiana, Springer*)

Regularity of the one-phase free boundaries (2020) - G. De Philippis - L. Spolaor - B. Velichkov (
*J. Eur. Math. Soc.*)

(Quasi-)conformal methods in two-dimensional free boundary problems (2024) - G. Buttazzo - F. P. Maiale - B. Velichkov (
*Rend. Lincei Mat. Appl.*)

Shape optimization problems in control form (2021) - F. P. Maiale - G. Tortone - B. Velichkov (
*Revista Matemática Iberoamericana*)

Epsilon-regularity for the solutions of a free boundary system (2023) - M. Colombo - L. Spolaor - B. Velichkov (
*Interfaces and Free Boundaries*)

Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem (2021) - N. Edelen - L. Spolaor - B. Velichkov (
*Indiana Univ. Math. J.*)

A strong maximum principle for minimizers of the one-phase Bernoulli problem (2024) - F. Générau - E. Oudet - B. Velichkov (
*Arch. Rat. Mech. Anal.*)

Cut locus on compact manifolds and uniform semiconcavity estimates for a variational inequality (2022) - N. Edelen - L. Spolaor - B. Velichkov (
*Calculus of Variations and Partial Differential Equations*)

The symmetric (log-)epiperimetric inequality and a decay-growth estimate (2023) - T. Schmidt (Accepted Paper:
*Math. Ann.*)

Isoperimetric conditions, lower semicontinuity, and existence results for perimeter functionals with measure data (2024) - G. P. Leonardi - G. Vianello (Accepted Paper:
*Transactions of the AMS*)

A Vertex-Skipping property for almost-minimizers of the relative perimeter in convex sets (2024) - G. Carlier - A. Figalli - F. Santambrogio (Accepted Paper:
*Ann. Inst. H. Poincaré Anal. Non Linéaire*)

On optimal transport maps between 1/d-concave densities (2024) - M. Datres - G. P. Leonardi - A. Figalli - D. Sutter (Accepted Paper:
*NeurIPS*)

A Two-Scale Complexity Measure for Deep Learning Models (2024) - S. Di Marino - A. Gerolin - L. Nenna (Accepted Paper:
*Letters in Mathematical Physics*)

Universal diagonal estimates for minimizers of the Levy-Lieb functional (2023) - F. Essebei - G. Giovannardi - S. Verzellesi (
*NoDEA Nonlinear Differential Equations Appl.*)

Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups (2024) - G. Giovannardi - A. Pinamonti - J. Pozuelo - S. Verzellesi (
*Adv. Math.*)

The prescribed mean curvature equation for $t$-graphs in the Sub-Finsler Heisenberg group $\mathbb{H}^n$ (2024) - S. Almi - M. Caponi - M. Friedrich - F. Solombrino (Accepted Paper:
*Mathematische Annalen*)

A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations (2024) - R. Marziani - F. Solombrino (
*Proceedings of the Royal Society Edinburgh Sect. A*)

Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation (2024) - F. Anceschi - G. Ascione - D. Castorina - F. Solombrino (
*Journal of Mathematical Analysis and Applications*)

Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift (2024) - F. Dragoni - Q. Liu - Y. Zhang (Preprint)

Horizontal semiconcavity for the square of Carnot-Carath\'eodory distance on step 2 Carnot groups and applications to Hamilton-Jacobi equations (2024) - G. Pascale - M. Pozzetta (
*Calculus of Variations and Partial Differential Equations*)

Quantitative isoperimetric inequalities for classical capillarity problems (2024) - F. Lagoutière - F. Santambrogio - S. Tran Tien (Accepted Paper:
*Journal de l'Ecole Polytechnique - Mathématiques*)

Vanishing viscosity limit for aggregation-diffusion equations (2024) - B. Han (Preprint)

Ricci tensor on smooth metric measure space with boundary (2017) - B. Han (
*Discrete and Continuous Dynamical Systems*)

Self-improvement of gradient estimate of heat flows on metric measure spaces (2018) - N. Gigli - B. Han (
*Journal of Functional Analysis*)

Sobolev Spaces on Warped Products (2018) - B. Han - K. T. Sturm (
*Annali di Matematica Pura ed Applicata*)

Curvature-dimension conditions for diffusions under time change (2022) - B. Han (
*Manuscripta Mathematica*)

The sharp $p$-Poincaré inequality under the measure contraction property (2020) - B. Han (
*Advances in Mathematics*)

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds (2020) - B. Han (
*Journal de Mathématiques Pures et Appliquées*)

Rigidity of some functional inequalities on ${\rm RCD}$ spaces (2021) - B. Han (
*The Journal of Geometric Analysis*)

Ricci tensor on ${\rm RCD}^*(K, N)$ spaces (2018) - B. Han (
*Potential Analysis*)

Conformal transformation on metric measure spaces (2019) - B. Han - E. Milman (
*Ann. Sc. Norm. Super. Pisa Cl. Sci.*)

Sharp Poincare inequality under Measure Contraction Property (2021) - B. Han (
*Calculus of Variations and Partial Differential Equations*)

Characterizations of monotonicity of vector fields on metric measure space (2018) - N. Apollonio - D. De Canditiis - G. Franzina - P. Stolfi - G. L. Torrisi (Accepted Paper:
*Stochastic Systems*)

Normal approximation of random Gaussian neural networks (2024) - A. Braides - E. Voglino - M. Zanardini (
*Netw. Heterogen. Media*)

Microstructures and anti-phase boundaries in long-range lattice systems (2024)