Renato Caccioppoli su MacTutor History of Mathematics archive

The Caccioppoli Prize is awarded every four years by the UMI to an Italian mathematician of established international reputation and who has not exceeded 38 years of age in the year preceding the award. Until 1970 the award was assigned every two years. The winner receives ten thousand euros and is nominated by a committee of five mathematicians appointed by the Main Office of the UMI.

## Winners

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

2014 Camillo De Lellis

### Caccioppoli Prize 1960 – Ennio De Giorgi (Scuola Normale Superiore di Pisa)

### Committee’s citation

Ennio De Giorgi’s high level scientific activity led him to gain full

professorship in 1958. Amongst his contributions, we mention, in

chronological order

- 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

The results obtained by Professor De Giorgi demonstrate his great talent

and put him in the front line of young Italian analysts.

Ennio De Giorgi on MacTutor History of Mathematics archive

### Caccioppoli Prize 1962 – Edoardo Vesentini (University of Pisa)

### Committee’s citation

Edoardo Vesentini’s high level scientific activity is mainly devoted to

Topology and to Complex Analysis, and in particular to the study of complex algebraic

varieties.

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)

### Caccioppoli Prize 1964 – Emilio Gagliardo (University of Genova)

### Committee’s citation

Gagliardo’s scientific production stands at very high levels, and gained

very high international recognition.

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)

### Caccioppoli Prize 1966 – Enrico Bombieri (University of Pisa)

### Committee’s citation

Enrico Bombieri’s scientific production is mainly concerned with Number

theory, Algebraic geometry and Analytic number theory, and includes

results of the highest level. It demonstrates Bombieri’s abilities to deal

with very modern tools of different kinds used to solve very difficult and

fundamental problems. His results have obtained clear international

recognition. Beside a few papers mainly dealing with the Geometry of

numbers, we mention: a group of papers devoted to Selberg’s formulas for

the estimation of rests in the “Prinzahlsatz” and their generalization to

a class of arithmetic functions, according to S.A Amitsur’s viewpoints;

yet, we mention a paper about Tu. V. Linnik’s “large sieve”. This last

one, in which a very ingenious Abel-type inequality on double sums is

employed, eventually leads to a theorem that in many applications can be

used to replace the extended Riemann hypothesis (see for instance a paper

on small differences of consecutive prime numbers). In the field of

Analytic functions, beside a few papers on meromorphic functions, he

mainly worked on the Bierberbach conjecture on univalent functions.

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

### Caccioppoli Prize 1968 – Mario Miranda (University of Pisa)

### Committee’s citation

Mario Miranda’s work, building on previous DeGiorgi’s one, led to

fundamental contributions to the theory of minimal surfaces.

### Caccioppoli Prize 1970 – Claudio Baiocchi (University of Pavia)

### Committee’s citation

Baiocchi is a high level mathematician who obtained, in a short time,

important and original results in various fields of Analysis, and, in

particular, on abstract differential equations and their applications to

evolutionary problems and in the Interpolation Theory between Banach

spaces. He also gave contributions to the problem of determining when the

“strong” and “weak” extensions of a differential operator coincide.

### Caccioppoli Prize 1974 – Alberto Tognoli (University of Pisa)

### Committee’s citation

In his wide scientific production Tognoli has systematically developed,

and has actually contributed to found, a modern theory of real analytic

spaces. This theory shows remarkable differences with that of complex

analytic spaces due to the appearance of completely new phenomena and

problems that Tognoli eventually overcame with his new methods. Amongst

his papers the one called “Su una congettura di Nash” stands out. In this

paper Tognoli shows how to give an answer to a fundamental approximation

problem by mean of cobordism theory methods.

### Caccioppoli Prize 1978 – Enrico Giusti (University of Pisa)

### Committee’s citation

After a few early papers on Schrödinger and wave equations, Giusti

devoted himself to the theory of nonlinear elliptic equations. His

research is mainly dealing with regularity problems for solutions to the

minimal surfaces equation, variational inequalities for minimal surfaces

with thin and discontinuous obstacles, partial regularity of solutions to

nonlinear elliptic systems. His result on the Bernstein problem, obtained

in collaboration with Bombieri and De Giorgi, is of exceptional

importance. His whole scientific production features truly remarkable

results and shows that Giusti gave important contributions to the

solution of difficult problems in nonlinear analysis.

### Caccioppoli Prize 1982 – Antonio Ambrosetti (International School for Advanced Studies, Trieste)

### Committee’s citation

The by now very wide scientific production of Prof. Antonio Ambrosetti is

mainly concerned with the following topics:

- 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

Such fields already gained contributions from several top

researchers. After entering them Ambrosetti was able to solve difficult

open problems (in particular, the one concerning the existence of periodic

solutions with prescribed minimal period) - 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

Professor Ambrosetti is a remarkably original researcher; when approaching

problems, rather than making use of massive technical tools he prefers

finding new and often surprising paths. At the origins of Ambrosetti’s

heuristic approaches we always find the attitude to catch the most basic

geometric and intuitive aspects which are typical of the Functional

Analysis “in the style of Caccioppoli”.

Antonio Ambrosetti on Wikipedia (italian)

### Caccioppoli Prize 1986 – Corrado De Concini (La Sapienza University of Rome)

### Committee’s citation

The committee proposes to award Prof. Corrado De Concini with the 1986

Caccioppoli prize for his contributions to the Theory of invariants, to

the Theory of algebraic groups, to Enumerative geometry, and, in

particular, for the solution (obtained with E. Arbarello) of the classical

Schottky’s problem.

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)

### Caccioppoli Prize 1990 – Gianni Dal Maso (International School for Advanced Studies, Trieste)

### Committee’s citation

Gianni Dal Maso is a very high level mathematician and one of the most

noteworthy members of the Italian school of Calculus of Variations. His

whole scientific production, which already includes more than sixty

papers, stands at very high levels; some of its highlights include

- 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

### Caccioppoli Prize 1994 – Nicola Fusco (University of Naples)

### Committee’s citation

Nicola Fusco is a refined mathematician who gave substantial contributions

to the Calculus of Variations. In his high level scientific production

some of the highlights are

- 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

### Caccioppoli Prize 1998 – Luigi Ambrosio (Scuola Normale Superiore di Pisa)

### Committee’s citation

Luigi Ambrosio is a remarkable mathematician who gave substantial

contributions to the Calculus of Variations and to some aspects of the

Theory of partial differential equations. In his very high level

scientific production, he obtained particularly relevant results on the

following topics:

- 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

### Caccioppoli Prize 2002 – Giovanni Alberti (University of Pisa)

### Committee’s citation

Giovanni Alberti is one of the most brilliant analysts of his generation.

He very soon revealed his talent with by proving a remarkable Lusin type

theorem for gradients, that eventually had important applications to

semicontinuity and relaxation problems for integral functionals. He

proved a deep structure theorem on the distributional derivative of BV

maps, which is nowadays known as Rank-one theorem, thereby solving in the

positive a conjecture of De Giorgi. He eventually studied subtle problems

in the theory of Gamma convergence and in the theory of multiple-scale

problems, and developed a general convergence theory for problems of

Ginzburg-Landau type in arbitrary dimension and codimension.

Giovanni Alberti on Wikipedia (italian)

### Caccioppoli Prize 2006 – Andrea Malchiodi (International School for Advanced Studies, Trieste)

### Committee’s citation

The work of Andrea Malchiodi is devoted to important problems of

variational nature, such as

- 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

On such topics, Malchiodi obtained several remarkable and original

results, that appeared on some of the most prestigious Mathematics

journals, and that gained international recognition. In particular, it is

worth mentioning his prestigious papers on Q-curvature and those on the

existence of solutions to singular perturbation problems concentrating on

positive dimension sets. Such results gained a wide international success,

making him one of the most brilliant figures in the nowadays Mathematical

community.

### Caccioppoli Prize 2010 – Giuseppe Mingione (University of Parma)

### Committee’s citation

The vast scientific production of Giuseppe Mingione, which is entirely

focused on regularity problems for vectorial integral functionals and

nonlinear systems of partial differential equations of elliptic and

parabolic type, is rich of remarkable results, that have been obtained by

mean of absolutely innovative techniques, in turn based on an original and

sometimes surprising way of using Potential theory ideas in the nonlinear

setting.

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

Above all, there must be mentioned two recent papers featuring pointwise

gradient estimates for solutions to quasilinear equations

via Riesz and Wolff potentials of the data. Such very general estimates

allow to obtain in a unified way several results proved in the last thirty

years by very different and peculiar techniques, and to fix some

borderline cases which were still remaining an open problem.

### Caccioppoli Prize 2014 – Camillo De Lellis, (Università di Zurigo)

### Committee’s citation

Camillo De Lellis è un matematico di grande talento e profondità che ha dato contributi importanti al Calcolo delle Variazioni, alla Teoria Geometrica della Misura, alla Fluidodinamica e alla teoria dei sistemi di leggi di conservazione. La sua si presenta come una figura di punta nel panorama internazionale dell’Analisi Matematica.

Nella sua produzione scientifica, tutta di altissimo livello e di ampio

spettro, spiccano

— I risultati, ottenuti con L. Székelyhidi, sull’esistenza di soluzioni anomale dell’equazione di Eulero. La loro costruzione, ottenuta con un originale uso dell’integrazione convessa di Gromov, si collega in modo importante alla teoria della turbolenza e permette di fare dei passi importanti verso la soluzione della congettura di Onsager.

— Una nuova dimostrazione della regolarità della regolarità delle soluzioni delle superficie minime in codimensione maggiore di uno, ottenuta con E. Spadaro.

— I risultati sui sistemi di leggi di conservazione. In particolare, quelli con L. Ambrosio e F. Bouchut sulla buona positura per il cosiddetto sistema di Keyfitz-Kranzer, e quelli con G. Crippa, in cui si sviluppa un approccio alternativo alla teoria di DiPerna-Lions per l’unicità delle equazioni di trasporto con coefficienti molto irregolari.

Camillo De Lellis su Wikipedia