2 J. MAWHIN

Problems of the calculus of variations had led Weierstrass to introduce a con-

cept of integral, widely generalized by Cesari in the form of a versatile interval

function, the Burkill-Cesari integral. With Domenico Candeloro, Patrizia Pucci

pushed further the study of this type of integrals, and of its relation with Riemann-

Stieltjes, Fubini-Tonelli, Serrin and Henstock-Kurzweil integrals. Alone, or in col-

laboration with Candeloro and with Giuseppe Vitillaro, Patrizia has contributed to

other questions related to the calculus of variations or optimal control, like lower

closure theorems, weak compactness conditions, du Bois-Reymond necessary condi-

tion, isoperimetric problems and Aumann integrals. Motivated by another aspect of

Cesari’s work, she has extended some well-posedness results to more general classes

of quasilinear hyperbolic systems. A singularity in Patrizia’s production during

this period is a result showing that the level surfaces of a classical solution of the

overdetermined Levi-Civita system Δu = f(u), |Du| = g(u) on a domain Ω ⊂ R3

must be pieces of concentric spheres, concentric cylinders, or parallel planes.

The strong potential of the young Italian mathematician revealed by those con-

tributions did not remain unoticed to some famous mathematicians in the United

States. The middle nineteen eighties saw the begining of a collaboration with

Lamberto Cesari and with James Serrin, which would last till their respective dis-

paritions in 1990 and 2012. Patrizia Pucci has, jointly or alone, contributed to

several favorite research topics of Cesari, like periodic solutions of nonlinear wave

equations, characterizations of weak convergence in L1 useful in optimization the-

ory, alternative method applied to nonlinear perturbations of non self-adjoint linear

operators, and bounded variation solutions of variational problems. After Cesari’s

death, Patrizia has been among the founding members, at the University of Perugia,

of the ‘Centro Studi Interfacolta Lamberto Cesari’ in 1995.

3. The exemplary collaboration with James Serrin (1984-2011)

The long, fruitful and inspiring collaboration with James Serrin started in 1984

with several papers related to Ambrosetti-Rabinowitz’s mountain pass lemma. As

it is well known, this useful result asserts that any C1-functional f on a Banach

space X, which satisfies a Palais-Smale compactness condition and is such that, for

some e ∈ X with e R and some a 0,

f(x) ≥ a 0 on ∂BR, f(0) a, f(e) a, (1)

has a critical point z with critical value f(z) = infγ∈Γ maxt∈[0,1] f(γ(t)) ≥ a, where

Γ denotes the set of all continuous path γ joining 0 to e. Pucci and Serrin have

answered a question of Rabinowitz by giving conditions under which z is a saddle

point, and have proved new variants of Ambrosetti-Rabinowitz’s conditions for the

existence of a critical point z which is not a local minimum, like

(1) 0 is a local minimum of f and f(e) f(0) for some e (in which case

f(z) ≥ f(0))

(2) 0 is a strict local minimum of f and f(e) ≤ f(0) for some e = 0 (in which

case f(z) f(0))

Furthermore, if 0 is a local minimum, either there is a critical point z which is not

a local minimum, or 0 is an absolute minimum and the set of absolute minima is

connected. When X is finite-dimensional, the last two inequalities in condition (1)

can be weakened to f(0) ≤ a, f(e) ≤ a and, for X infinite-dimensional, condition