CACCIOPPOLI PRIZE
UMI - Italian Mathematical Union |

1960 Ennio De Giorgi

1962 Edoardo Vesentini

1964 Emilio Gagliardo

1966 Enrico Bombieri

1968 Mario Miranda

1970 Claudio Baiocchi

1974 Alberto Tognoli

1978 Enrico Giusti

1982 Antonio Ambrosetti

1986 Corrado De Concini

1990 Gianni Dal Maso

1994 Nicola Fusco

1998 Luigi Ambrosio

2002 Giovanni Alberti

2006 Andrea Malchiodi

2010 Giuseppe Mingione

1962 Edoardo Vesentini

1964 Emilio Gagliardo

1966 Enrico Bombieri

1968 Mario Miranda

1970 Claudio Baiocchi

1974 Alberto Tognoli

1978 Enrico Giusti

1982 Antonio Ambrosetti

1986 Corrado De Concini

1990 Gianni Dal Maso

1994 Nicola Fusco

1998 Luigi Ambrosio

2002 Giovanni Alberti

2006 Andrea Malchiodi

2010 Giuseppe Mingione

- His papers on the Cauchy problem, featuring a new and ingenious technique which is applied to prove both an existence theorem and nonexistence results
- His exceptionally interesting works on the notion of sets of finite perimeter in which, building on previous contributions of Caccioppoli, he obtained new results that in turn allowed him to prove the isoperimetric property of the sphere in the same class of finite perimeter sets
- A very deep paper on the analytic character of the extremals of multiple integrals, which is dedicated to the study of a very longstanding issue, that has gained international recognition and that is the starting point for several subsequent works on elliptic equations

Ennio De Giorgi on MacTutor History of Mathematics archive

photo courtesy of MFO

After his first publications (amongst which we mention one paper dedicated to the study of polarity of algebraic curves and another one where an intrinsic proof of Riemann-Roch theorem is obtained), there must be recalled his proof of a fundamental theorem connecting, via duality, the homology classes of canonical varieties of an algebraic variety with the Chern's cohomology classes, and his deep papers on jacobians of several meromorphic functions on a complex variety, that generalize the classical results related to algebraic varieties.

In the last years Vesentini started a fruitful cooperation with other brilliant mathematicians that eventually led to very interesting results on non-compact complex varieties related to Kodaira's embedding theorem.

The brilliant and deep work of Vesentini, demonstrating a masterly use of modern topological, geometrical and analytical methods, and the importance of the results he achieved, clearly qualify him as one of the best young Italian mathematicians.

Edoardo Vesentini on Wikipedia (Italian)

After some early, interesting contributions, amongst which we single out those on second order parabolic equations, he obtained, under the guidance of Caccioppoli, a compactness criterion with respect to mean convergence. In turn, this led him to give a characterization of boundary traces that was the starting point for the work of several authors, especially from Russia.

Further remarkable results of Gagliardo are those on Sobolev spaces, including some generalizations of a few inequalities due to Sobolev himslef. Gagliardo obtained important results in Interpolation Theory. In particular, he was the first to build an interpolation theory for Banach spaces. Further, more recent results of Gagliardo are concerned with integral transformations between Lebesgue spaces. The results of Gagliardo show that he is a very original analyst, and obviously put him amongst the best young Italian mathematicians.

Emilio Gagliardo on Wikipedia (Italian)

As for his work in Algebraic geometry, we mention two particularly remarkable papers: a first one dealing with the exponential sums related to the L-function introduced by Artin and Schreirer, and another one (written with Swinnerton-Dyer) on the Artin's Z-function of the cubic hypersurface in the four dimensional projective space. For this case it is possible to verify the validity of the Riemann-Weil's conjecture.

Enrico Bombieri on MacTutor History of Mathematics archive

photo courtesy of MFO

photo courtesy of MFO

photo courtesy of MFO

- Invertibility of maps between Banach spaces by mean of geometric methods
- Critical point theory for functionals, i.e. "Calculus of Variations in
the large". As for this topic - which is one of those Ambrosetti mostly
studied - it is possible to make a more detailed classification:
- Critical points problems on manifolds with boundary
- Eigenvalue problems for nonlinear operators and existence problems with infinitely many critical points
- Periodic solutions to Hamiltonian systems

- Nonlinear problems with discontinuities (a very promising field Ambrosetti recently devoted an interesting paper to)
- Various issues motivated by applications: plasma equilibria, existence of vortices, plate equilibria

Antonio Ambrosetti on Wikipedia (Italian)

Such works show that De Concini is a very high level mathematician, who is able to use with great skills algebraic, topological and analytical methods towards the solution of difficult problems.

Corrado De Concini on Wikipedia (Italian)

- The results on the asymptotic behavior of sequences of variational obstacle problems
- The results on linear and quasilinear Dirichlet problems in variable domains with rough boundary (behavior of solutions with respect to the domain, determination of the asyptotic problem via the method of asymptotic capacities, Wiener criterion, 1985)
- The results on functionals defined on the space of Special Functions of Bounded Variation (SBV) and related interesting applications to image segmentation problems

- The lower semicontinuity theorem for quasiconvex functionals with respect to the natural topology, that closes a research field many authors had previously contributed to
- The partial regularity theorems, extending previously known results given by several different authors and elegantly avoiding some unnecessary assumptions which were usually considered in the literature
- The recent papers on the regularity of minimizers of the Mumford-Shah functional, allowing to draw a complete picture of the image recognition problem

- Existence, regularity and approximation problems in the theory of free discontinuity problems
- Semicontinuity and relaxation for integral functionals
- Singularities of convex functions
- Curvature flow of manifolds in arbitrary codimension

Giovanni Alberti on Wikipedia (Italian)

- The Yamabe problem and the scalar curvature problem
- Conformal geometric problems for fourth order operators
- Concentration phenomena for singular perturbation problems
- Solitary waves for Schrödinger nonlinear equations and related semiclassical states

Some of the highlights are

- The singular sets estimates for minima of vectorial integral functionals, which close an entire set of problems that had remained open since the seventies
- The boundary regularity results for non-linear systems of p-Laplacean type
- The Calderón-Zygmund type results on the existence of higher order fractional derivatives of solutions to measure data problems

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