Giovanni Alberti
Research


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SUMMARY

research and cv
links
preprints
recent or most relevant papers
survey papers
some talks and lecture notes
all papers (archive page)


RESEARCH & CV    [back to top]

I have been working in the following areas: Real Analysis, Geometric Measure Theory and Calculus of Variations, with applications to partial differential equations and continuum mechanics (capillarity, pattern formation, phase separation, Ginzburg-Landau vortices).

cv in english     cv in italian

I am currently involved in the following research projects:

Calcolo delle variazioni
 [Calculus of variations], PRIN project directed by Luigi Ambrosio (Scuola Normale Superiore, Pisa, Italy)

Local structure of sets, measures and currents, funded by an ERC advanced grant awarded to David Preiss (University of Warwick, UK)

Fluid flows and irregular transport, funded by an ERC starting grant awarded to Gianluca Crippa (University of Basel, Switzerland)


LINKS    [back to top]
CVGMT preprint server
Department of Mathematics, University of Pisa
Centro di ricerca matematica E. De Giorgi, Pisa


WARNING
For each paper below, 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").


PREPRINTS    [back to top]
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RECENT OR MOST RELEVANT PAPERS    [back to top]

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]

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]

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 Lp-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]

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]

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]

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]

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]

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)

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]

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 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]

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]

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]

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


SURVEY PAPERS    [back to top]

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]

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]

Geometric measure theory
in the "Encyclopedia of mathematical physics", vol. 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]

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]

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]"

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]


SOME TALKS AND LECTURE NOTES    [back to top]

Quasistatic evolution and contact-angle hysteresis in capillarity
course in the "ITN-Winterschool on Mathematical Models for Wetting: Analysis and Numerics"
Veilbronn (near Erlangen), February 13-17, 2012
[abstract.txt  -  lect1.pdf (4.5 Mb)  -  lect2.pdf (5.7 Mb)  -  lect3.pdf (5.1 Mb)  -  lect4.pdf (5.5 Mb)]

Classical and non-classical tools for minimal surfaces
course in the "Fifth Summer School in Analysis and Applied Mathematics"
Roma, June 1-5, 2009
[abstract.txt  -  lect1.pdf (1.8 Mb)  -  lect2.pdf (2.1 Mb)  -  lect3.pdf (2.1 Mb)  -  lect4.pdf (1.4 Mb)]

Sulla struttura degli insiemi di misura nulla. Applicazioni e problemi aperti
plenary lecture at the "XVIII congresso dell'Unione Matematica Italiana"
Bari, 24-29 settembre 2007
[abstract.txt  -  complete presentation.pdf (1.9 Mb)]

Structure of null sets in the Euclidean space. Some results and open problems
lecture at the conference "Real Analysis, Geometric Measure Theory, PDE and Banach Spaces" University of Warwick, August 17-19, 2007
[abstract.txt  -  complete presentation.pdf (2.3 Mb)]

On maps whose distributional Jacobian is a measure
lecture at the "Summer Symposium in Real Analysis XXXI"
Trinity College, Oxford, August 12-16, 2007
[abstract.txt  -  complete presentation.pdf (1.9 Mb)]

Some applications of geometric measure theory
course in the school: "Geometric Measure Theory: Old and New"
Les Diablerets, April 3-8, 2005
[abstract.txt  -  complete presentation.pdf (2.8 Mb)]