Giovanni Alberti
Archive of mathematical papers
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Warning
This page contains all my papers and research notes, starting from the most recent one.
For every paper, the version stored here is as close as possible to the published one, but may be not exactly the same. In a few cases I also kept versions which are different from the published one, for example because they contain parts that do not appear in the published version.
Most papers are also available in a paper-saving format (two pages per side, suffix "rdc").

Integration of nonsmooth 2-forms: from Young to Itô and Stratonovich
with Eugene Stepanov and Dario Trevisan
J. Funct. Anal., 286 (2024), no. 2, art. 110212
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

On the geometric structure of currents tangent to smooth distributions
with Annalisa Massaccesi and Eugene Stepanov
J. Differential Geom., 122 (2022), no. 1, 1-33
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Lossless analog compression
with Helmut Bölcskei, Camillo De Lellis, Gunther Koliander and Edwin Riegler
IEEE Trans. Inform. Theory, 65 (2019), no. 11, 7480-7513
[full text.pdf  -  MathSciNet  -  DOI]

Optimal reinforcing networks for elastic membranes
with Giuseppe Buttazzo, Serena Guarino Lo Bianco, and Édouard Oudet
Netw. Heterog. Media, 14 (2019), no. 3, 589-615
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Loss of regularity for the continuity equation with non-Lipschitz velocity field
with Gianluca Crippa and Anna L. Mazzucato
Ann. PDE, 5 (2019), no. 1, art. 9
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Exponential self-similar mixing by incompressible flows
with Gianluca Crippa and Anna L. Mazzucato
J. Amer. Math. Soc. (JAMS), 32 (2019), no. 2, 445-490
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

On some geometric properties of currents and Frobenius theorem
with Annalisa Massaccesi
Rend. Lincei Mat. Appl., 28 (2017), no. 4, 861-869
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

On the structure of continua with finite length and Gołąb's semicontinuity theorem
with Martino Ottolini
Nonlinear Anal. 153 (2017), 35-55
[postprint.pdf  -  postprint-rdc.pdf  -  revised.pdf  -  revised-rdc.pdf  -  MathSciNet  -  DOI]
Note: compared to the postprint version (which is essentially the same as the published one), in the revised version we have added two pictures, corrected the proof of Lemma 2.22 and a few small mistakes, and modified the definition of delta-partition in Subsection 2.7, in order to make the statement of Proposition 2.8 stronger.

Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux. I
with Stefano Bianchini and Laura Caravenna
J. Differential Equations, 261 (2016), no. 8, 4298-4337
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Lossless linear analog compression
with Helmut Bölcskei, Camillo De Lellis, Günther Koliander, Erwin Riegler
in "Proceedings of the IEEE International Symposium on Information Theory", volume 2016, pp. 2789-2793
Institute of Electrical and Electronics Engineers, New York, 2016
[full text.pdf  -  DOI]

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space
with Andrea Marchese
Geom. Funct. Anal. (GAFA), 26 (2016), no. 1, 1-66
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Reduction on characteristics for continuous solutions of a scalar balance law
with Stefano Bianchini and Laura Caravenna
in "Hyperbolic problems: theory, numerics, applications" (Proceedings of the 14th International Conference on Hyperbolic Problems (HYP2012), Padua, June 25-29, 2012), part III (contributed talks), pp. 399-406.
Edited by F. Ancona, A. Bressan,  P. Marcati, A. Marson.
AIMS Series on Applied Math., volume 8.
American Institute of Mathematical Science, Springfield, 2014
[full text.pdf  -  full text-rdc.pdf  -  complete volume  -  ISBN: 978-1-60133-017-8]

Exponential self-similar mixing and loss of regularity for continuity equations
with Gianluca Crippa and Anna L. Mazzucato
C. R. Acad. Sci. Paris, Ser. I, 352 (2014), no. 11, 901-906
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

On the L^p-differentiability of certain classes of functions
with Stefano Bianchini and Gianluca Crippa
Rev. Mat. Iberoam., 30 (2014), no. 1, 349-367
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

A uniqueness result for the continuity equation in two dimensions
with Stefano Bianchini and Gianluca Crippa
J. Eur. Math. Soc. (JEMS), 16 (2014), no. 2, 201-234
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Structure of level sets and Sard-type properties of Lipschitz maps
with Stefano Bianchini and Gianluca Crippa
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 12 (2013), no. 4, 863-902
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Generalized N-property and Sard theorem for Sobolev maps
Rend. Lincei Mat. Appl., 23 (2012), no. 1, 477-491
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Quasistatic evolution of sessile drops and contact angle hysteresis
with Antonio DeSimone
Arch. Rational Mech. Anal., 202 (2011), no. 1, 295-348
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Differentiability of Lipschitz functions, structure of null sets, and other problems
with Marianna Csörnyei and David Preiss (lecture delivered by M. Csörnyei)
in "Proceedings of the international congress of mathematicians (ICM 2010, Hyderabad, India, August 19-27, 2010)", volume 3 (invited lectures), pp. 1379-1394.
Edited by R. Bhatia et al.
Hindustan Book Agency, New Delhi, 2010
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI  -  ISBN: 978-981-4324-30-4]

Two-dimensional transport equation with Hamiltonian vector fields
with Stefano Bianchini and Gianluca Crippa
in "Hyperbolic problems: theory, numerics and applications" (proceedings of the 12th International Conference on Hyperbolic Problems (HYP2008), University of Maryland, June 9-13, 2008), part 2 (contributed talks), pp. 337-346.
Edited by E. Tadmor, J.-G. Liu, A. Tzavaras.
Proceedings of Symposia in Applied Mathematics, 67.
American Mathematical Society, Providence 2009
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI  -  ISBN: 978-0-8218-4730-5]

Uniform energy distribution for an isoperimetric problem with long-range interactions
with Rustum Choksi and Felix Otto
J. Amer. Math. Soc., 22 (2009), no. 2, 569-605
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Calcolo delle variazioni [Calculus of Variations]
in:  "Enciclopedia italiana di scienze, lettere ed arti. XXI Secolo. Settima Appendice", volume 1, pp. 252-255.
Edited by T. Gregory et al.
Istituto della Enciclopedia Italiana fondata da Giovanni Treccani, Roma 2006
[full text.pdf]

On maps whose distributional Jacobian is a measure
Real analysis exchange. Report on the "Summer Symposium in Real Analysis XXXI" (Trinity College, Oxford, England, August 13-15, 2007), pp. 153-162.
Michigan State University Press, 2008
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet]

Geometric measure theory
in the "Encyclopedia of mathematical physics", volume 2, pp. 520-527.
Edited by J.-P. Françoise, G.L. Naber, S.T. Tsou.
Academic Press / Elsevier Science, Oxford 2006
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI  -  ISBN: 978-0-12-512666-3]

Distributional Jacobian and singularities of Sobolev maps
Ricerche Mat., 54 (2005), no. 2, 375-394
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet]

Variational convergence for functionals of Ginzburg-Landau type
with Sisto Baldo and Giandomenico Orlandi
Indiana Univ. Math. J. 54 (2005), no. 5, 1411-1472
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Structure of null sets in the plane and applications
with Marianna Csörnyei and David Preiss (lecture delivered by G. Alberti)
in "European Congress of Mathematics. Proceedings of the 4th Congress (4ECM) held in Stockholm,
June 27-July 2, 2004", pp. 3-22.
Edited by A. Laptev.
European Mathematical Society (EMS), Zürich 2005
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI  -  ISBN 978-3-03719-009-8]

BV has the bounded approximation property
with Marianna Csörnyei, Aleksander Pełczyński and David Preiss
J. Geom. Anal.  15 (2005), no. 1, 1-7
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Wetting of rough surfaces: a homogenization approach
with Antonio DeSimone
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2053, 79-97
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Aritmetica finita e crittografia a chiave pubblica. Un percorso didattico per studenti delle scuole medie superiori [Arithmetic and public-key cryptography. An educational journey for high-school students]
in "Ricordando Franco Conti", pp. 1-29.
Edited by A. Abbondandolo, M. Giaquinta, F. Ricci.
Pubblicazioni del Centro di Ricerca Matematica E. De Giorgi, Scuola Normale Superiore, Pisa 2004
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  ISBN: 978-8876421372]
Note: a preliminary version of these notes appeared in 2001 under the title of "Aritmetica e crittografia [Arithmetic and cryptography]"

Functions with prescribed singularities
with Sisto Baldo and Giandomenico Orlandi
J. Eur. Math. Soc. (JEMS) 5 (2003), no. 3, 275-311
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

The calibration method for the Mumford-Shah functional and free discontinuity problems
with Guy Bouchitté and Gianni Dal Maso
Calc. Var. Partial Differential Equations, 16 (2003), no. 3, 299-333
[postprint.pdf  -  postprint-rdc.pdf  -  extended.pdf  -  extended-rdc.pdf  -  MathSciNet  -  DOI]
Note: the extended version contains some remarks and paragraphs (not numbered) not included in the postprint version (which is essentially the same as the published one)

Un risultato di convergenza variazionale per funzionali di tipo Ginzburg-Landau in dimensione qualunque
[A variational convergence result for functionals of Ginzburg-Landau type in any dimension]
Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 4 (2001), no. 2, 289-310
[ full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  EuDML]
[English translation (unpublished): full text.pdf  -  full text-rdc.pdf]

Denjoy-Young-Saks theorem for approximate derivatives revisited
with Marianna Csörnyei, Miklós Laczkovich and David Preiss
Real Anal. Exchange, 26 (2000/01), no. 1, 485-488
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  Project Euclid]

On a long standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property
with Luigi Ambrosio and Xavier Cabré
Acta Appl. Math., 65 (2001), no. 1-3, 9-33
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

A new approach to variational problems with multiple scales
with Stefan Müller
Comm. Pure Appl. Math., 54 (2001), no. 7, 761-825
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Some remarks about a notion of rearrangement
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), no. 2, 457-472
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  Numdam  -  EuDML]

Variational models for phase transitions, an approach via Gamma-convergence
in L. Ambrosio, N. Dancer: "Calculus of variations and partial differential equations. Topics on geometrical evolution problems and degree theory. Papers from the Summer School held in Pisa, September 1996", pp. 95-114.
Edited by G. Buttazzo, A. Marino, M.K.V. Murthy.
Springer-Verlag, Berlin, 2000
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI  -  ISBN 978-3-540-64803-1]

The calibration method for the Mumford-Shah functional
with Guy Bouchitté and Gianni Dal Maso
C. R. Acad. Sci. Paris, Sér. I Math., 329 (1999), no. 3, 249-254
[full text.pdf  -  MathSciNet  -  DOI]

A geometrical approach to monotone functions in Rⁿ
with Luigi Ambrosio
Math. Z., 230 (1999), no. 2, 259-316
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Phase transition with line-tension effect
with Guy Bouchitté and Pierre Seppecher
Arch. Rational Mech. Anal., 144 (1998), no. 1, 1-46
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

A nonlocal anisotropic model for phase transitions: asymptotic behaviour of rescaled energies
with Giovanni Bellettini
European J. Appl. Math., 9 (1998), no. 3, 261-284
[postprint.pdf  -  postprint-rdc.pdf  -  extended.pdf  -  extended-rdc.pdf  -  MathSciNet  -  DOI]
Note: compared to the postprint version (which is essentially the same as the published one) the extended version contains an additional chapter where we prove the convergence of certain nonlocal interaction energies to standard convex functionals of the gradients

A nonlocal anisotropic model for phase transitions. I. The optimal profile problem
with Giovanni Bellettini
Math. Ann., 310 (1998), no. 3, 527-560
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

A note about the theory of SBV functions
with Carlo Mantegazza
Boll. Un. Mat. It. B (7), 11 (1997), no. 2, 375-382
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet]

Surface tension in Ising systems with Kac potential
with Giovanni Bellettini, Marzio Cassandro and Errico Presutti
J. Statist. Phys., 82 (1996), no. 3-4, 743-796
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Boundary effects in phase transitions
with Guy Bouchitté and Pierre Seppecher
in "Curvature flows and related topics. Proceedings of the Second International Conference held in Levico, June 27-July 2, 1994", pp. 1-11.
Edited by A. Damlamian, J. Spruck, A. Visintin.
Gakuto International Series, Mathematical Sciences and Applications, 5.
Gakkotosho, Tokyo, 1995
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  ISBN: 978-4762504143]

Un résultat de perturbations singuliéres avec la norme H^1/2
with Guy Bouchitté and Pierre Seppecher
C.R. Acad. Sci. Paris, Sér. I Math., 319 (1994), no. 4, 333-338
[full text.pdf  -  MathSciNet  -  Gallica]

Non-occurrence of gap for one-dimensional autonomous functionals
with Francesco Serra Cassano
in "Calculus of variations, homogenization and continuum mechanics. Proceedings of the workshop held in Marseille, June 21-25, 1993", pp. 1-17.
Edited by G. Bouchitté, G. Buttazzo, P. Suquet.
Series on Advances in Mathematics for Applied Sciences, 18.
World Scientific Publishing Co., River Edge, NJ, 1994
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  ISBN: 978-9810217839]

Gap phenomenon for autonomous functionals
with Pietro Majer
J. Convex Anal., 1 (1994), no. 1, 31-45
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  EuDML]

On the structure of singular sets of convex functions
Calc. Var. Partial Differential Equations, 2 (1994), no. 1, 17-27
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Singular perturbation problems with a compact support semilinear term
with Luigi Ambrosio and Giuseppe Buttazzo
Asymptotic Anal., 8 (1994), no. 3, 247-258
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Local mappings on spaces of differentiable functions
with Giuseppe Buttazzo
Manuscripta Math., 79 (1993), no. 1, 81-97. Errata, ibid., 81 (1993), no. 3-4, 445-446
[corrected version.pdf  -  corrected version-rdc.pdf  -  MathSciNet  -  DOI]
Note: the version stored here includes all corrections from the erratum

Rank one property for derivatives of functions with bounded variation
Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), no. 2, 239-274
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Integral representation of local functionals
Ann. Mat. Pura Appl. (4), 165 (1993), no. 1, 49-86
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

On the singularities of convex functions
with Luigi Ambrosio and Piermarco Cannarsa
Manuscripta Math., 76 (1992), no. 3-4, 421-435
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]

Integral representation of functionals defined on Sobolev spaces
with Giuseppe Buttazzo
in "Composite media and homogenization theory. Proceedings of the workshop held in Trieste, January 15-26, 1990", pp. 1-12.
Edited by G. Dal Maso, G. Dell'Antonio.
Progress in Nonlinear Differential Equations and their Applications, 5.
Birkhäuser Boston, Boston 1991
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  ISBN: 978-1-4684-6787-1]

A Lusin type theorem for gradients
J. Funct. Anal., 100 (1991), no. 1, 110-119
[full text.pdf  -  full text-rdc.pdf  -  MathSciNet  -  DOI]