Gaetano Fichera

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Gaetano Fichera
Università di Roma
, 1941
Known for
Awards
Scientific career
FieldsMathematics
Institutions
  • Istituto Nazionale di Alta Matematica
  • Istituto Nazionale per le Applicazioni del Calcolo
  • Università di Trieste
  • Università di Roma "La Sapienza"
Doctoral advisorMauro Picone
Doctoral studentsMaria Adelaide Sneider

Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian

several complex variables. He was born in Acireale, and died in Rome
.

Biography

He was born in

Nazist troops, kept imprisoned in Teramo and then sent to Verona: he succeeded in escaping from there and reached the Italian region of Emilia-Romagna, spending with partisans the last year of war. After the war he was first in Rome and then in Trieste
, where he met Matelda Colautti, who became his wife in 1952.

Education and academic career

After graduating from the

Accademia Nazionale dei Lincei and first director of the journal Rendiconti Lincei – Matematica e Applicazioni,[3] he succeeded in reviving its reputation.[4]

Honours

He was member of several

Russian Academy of Science
.

Teachers

His lifelong friendship with his teacher

holomorphic function of several variables, as Fichera (1957, p. 707) recalls: the result was the paper (Fichera 1957), which is a masterpiece, although not generally acknowledged for various reasons described by Range (2002, pp. 6–11). Other scientists he had as teachers during the period 1939–1941 were Enrico Bompiani, Leonida Tonelli and Giuseppe Armellini: he remembered them with great respect and admiration, even if he did not share all their opinions and ideas, as Colautti Fichera (2006
, p. 16) recalls.

Friends

A complete list of Fichera's friends includes some of the best scientists and

Istituto Nazionale per le Applicazioni del Calcolo: the machine they advised to purchase was the first computer ever working in Italy. The most complete source about his friends and collaborators is the book (Colautti Fichera 2006
) by his wife Matelda: in those reference it is also possible to find a fairly complete description of Gaetano Fichera's scientific journeys.

The close friendship between Angelo Pescarini and Fichera has not his roots in their scientific interests: it is another war story. As Oleinik (1997, p. 12) recalls, Gaetano, being escaped from Verona and hidden in a convent in Alfonsine, tried to get in touch with the local group of partisans in order to help the people of that town who had been so helpful with him: they were informed about an assistant professor to the chair of higher analysis in Rome who was trying to reach them. Angelo, which was a student of mathematics at the University of Bologna under Gianfranco Cimmino, a former pupil of Mauro Picone, was charged of the task of testing the truth of Gaetano's assertions, examining him in mathematics: his question was:– "Mi sai dire una condizione sufficiente per scambiare un limite con un integrale (Can you give me a sufficient condition for interchanging limit and integration)?"–. Gaetano quickly answered:– "Non solo ti darò la condizione sufficiente, ma ti darò anche la condizione necessaria e pure per insiemi non-limitati (I can give you not only a sufficient condition, but also a necessary condition, and not only for bounded domains, but also for unbounded domains)"–. In effect, Fichera proved such a theorem in the paper (Fichera 1943), his latest paper written in while he was in Rome before joining the army: from that moment on he often used to joke saying that good mathematicians can always have a good application, even for saving one's life.

One of his best friends and appreciated scientific collaborator was

Olga Arsenievna Oleinik: she cured the redaction of his last posthumous paper (Fichera 1997), as Colautti Fichera (2006
, pp. 202–204) recalls. Also, she used to discuss his work with Gaetano, as he did with her: sometimes their discussion become lively, but nothing more, since they were extremely good friends and estimators of each one's work.

Work

Research activity

He is the author of more than 250 papers and 18 books (monographs and course notes): his work concerns mainly the fields of pure and applied mathematics listed below. A common characteristic to all of his research is the use of the methods of functional analysis to prove existence, uniqueness and approximation theorems for the various problems he studied, and also a high consideration of the analytic problems related to problems in applied mathematics.

Mathematical theory of elasticity

His work in

smooth boundary. Also he is known for his researches in the theory of hereditary elasticity: the paper (Fichera 1979b) emphasizes the necessity of analyzing very well the constitutive equations of materials with memory in order to introduce models where an existence and uniqueness theorems can be proved in such a way that the proof does not rely on an implicit choice of the topology of the function space where the problem is studied. At last, it is worth to mention that Clifford Truesdell invited him to write the contributions (Fichera 1972a) and (Fichera 1972b) for Siegfried Flügge
's Handbuch der Physik.

Partial differential equations

He was one of the pioneers in the development of the abstract approach through

Lax–Milgram theorem. He studied deeply the mixed boundary value problem i.e. a boundary value problem where the boundary has to satisfy a mixed boundary condition: in his first paper on the topic, (Fichera 1949), he proves the first existence theorem for the mixed boundary problem for self-adjoint operators of n > 2 variables, while in the paper (Fichera 1955a, pp. 22–29) he proves the same theorem dropping the hypothesis of self-adjointness. He is, according to Oleinik (1997), the founder of the theory of partial differential equations of non-positive characteristics: in the paper (Fichera 1956) he introduced the now called Fichera's function, in order to identify subsets of the boundary of the domain where the boundary value problem for such kind of equations is posed, where it is necessary or not to specify the boundary condition: another account of the theory can be found in the paper (Fichera 1960), which is written in English and was later translated in Russian and Hungarian.[9]

Calculus of variation

His contributions to the

partial differential operators
.

Functional analysis and eigenvalue theory

Main article: Fichera's existence principle

It is difficult to single out his contributions to functional analysis since, as stated at the beginning of this section, the methods of functional analysis are ubiquitous in his research: however, it is worth to remember paper (Fichera 1955a), where an important existence theorem is proved.[10]

His contributions in the field of eigenvalue theory began with the paper (Fichera 1955b), where he formalizes a method developed by Mauro Picone for the approximation of eigenvalues of operators subject only to the condition that their inverse is compact: however, as he admits in (Fichera 1974a, pp. 13–14), this method does not give any estimate on the approximation error on the value of the calculated (approximated) eigenvalues.

He contributed also to the classical

symmetric operators, introducing the method of orthogonal invariants.[11]

Approximation theory

His work in this field is mainly related to the study of systems of

poles, simple or not. The paper (Fichera 1974b) surveys the contribution to the solution of this and related problems by Sergey Mergelyan, Lennart Carleson, Gábor Szegő
as well as others, including his own.

Potential theory

His contributions to

V.G. Maz'ya, S.A. Nazarov, B.A. Plamenevsky, B.W. Schulze
and others testify) can be included in between his works in potential theory.

Measure and integration theory

His main contributions to those topics and are the papers (

minimizing a particular functional
.

Complex analysis of functions of one and several variables

He contributed to both the classical topic of

functions of several complex variables, i.e. pluriharmonic functions: starting from the paper (Amoroso 1912) he gives a trace condition analogous to the tangential Cauchy–Riemann condition for the solvability of the Dirichlet problem for pluriharmonic functions in the paper (Fichera 1982a), and generalizes a theorem of Luigi Amoroso to the complex vector space
for n ≥ 2
smooth domain by Luigi Amoroso in his cited paper, the Amoroso integro-differential equation, is a necessary and sufficient condition for the solvability of the Dirichlet problem for pluriharmonic functions when this domain is the sphere
in .[17]

Exterior differential forms

His contributions to the theory of

axiomatic method
, proving an abstract form of Hodge theorem.

Numerical analysis

As noted in the "

biological sciences.[21][22]

History of mathematics

his work in this field occupy all the volume (Fichera 2002). He wrote bibliographical sketches for a number of mathematicians, both teachers, friends and collaborators, including Mauro Picone, Luigi Fantappiè, Pia Nalli, Maria Adelaide Sneider, Renato Caccioppoli, Solomon Mikhlin, Francesco Tricomi, Alexander Weinstein, Aldo Ghizzetti. His historical works contain several observations against the so-called historical revisitation: the meaning of this concept is clearly stated in the paper (Fichera 1996). He identifies with the word revisitation the analysis of historical facts basing only on modern conceptions and points of view: this kind of analysis differs from the "true" historical one since it is heavily affected by the historian's point of view. The historian applying this kind of methodology to history of mathematics, and more generally to the history of science, emphasizes the sources that have led a field to its modern shape, neglecting the efforts of the pioneers.

Selected publications

A selection of Gaetano Fichera's works was published respectively by the

Unione Matematica Italiana and the Accademia Pontaniana in his "opere scelte" (Fichera 2004) and in the volume (Fichera 2002). These two references include most of the papers listed in this section: however, these volumes does not include his monographs and textbooks
, as well as several survey papers on various topic pertaining to his fields of research.

Papers

Research papers

Historical and survey papers

Monographs and textbooks

See also

Notes

  1. ^ The main reference about his personal life is the book (Colautti Fichera 2006).
  2. ^ His last lesson of the course of higher analysis was published in (Fichera 1995a).
  3. Accademia Nazionale dei Lincei
    .
  4. ^ See Colautti Fichera (1997, p. 14, footnote), and Galletto (2007, p. 142).
  5. ^ The episode is narrated in (Colautti Fichera 2006, pp. 30–31).
  6. ^ See also its English translation (Fichera 1964b).
  7. ^ These are his only papers in the field of variational inequalities: see the article "Signorini problem" for a discussion of the reasons why he left this field of research.
  8. ^ The same paper was previously published in Russian in a volume in honour of Ilia Vekua: see Colautti Fichera (1997, p. 29) for the exact reference.
  9. ^ See the bibliography (Colautti Fichera 1997): some of the translated papers are available online from the All-Russian Mathematical Portal.
  10. ^ This is Fichera's existence principle: see the survey paper by Valent (1999, p. 84).
  11. ^ See (Fichera 1974a, pp. 33–127), (Fichera 1978a), (Weinberger 1999) and references therein.
  12. ^ See also the monograph (Günther 1967).
  13. ^ See also the "Approximation theory" section.
  14. ^ See the paper (Range 2002).
  15. ^ Introduced by him in the same paper.
  16. ^ See also (Fichera 1986), where the theorem is presented in English and extended to the case that the normal vector and the Dirichlet boundary condition are only continuous.
  17. ^ The details can be found in the paper (Fichera 1982c).
  18. ^ He tells this story in his last lesson (Fichera 1995a, pp. 18–19): see also (Colautti Fichera 2006, p. 21).
  19. sheaf theory
    shows.
  20. ^ See also the recollections of Wendland in (Wendland 2007, p. 8).
  21. ^ See also the research announcement (Fichera, Sneider & Wyman 1977a),
  22. ^ Note that Oleinik (1993, pp. 12–13) describes it as a work in the theory of ordinary differential equations, perhaps reflecting the difficulty of classifying such kind of research.
  23. ^ See (Günther 1967, §24) where the results of this paper are reported.

References

Biographical references

General references

Scientific references

Publications dedicated to him or to his memory

External links
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