Gaetano Fichera


Gaetano Fichera was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.

Biography

He was born in Acireale, a town near Catania in Sicily, the elder of the four sons of Giuseppe Fichera and Marianna Abate. His father Giuseppe was a professor of mathematics and influenced the young Gaetano starting his lifelong passion. In his young years he was a talented football player. On 1 February 1943 he was in the Italian Army and during the events of September 1943 he was taken prisoner by the 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 liceo classico in only two years, he entered the University of Catania at the age of 16, being there from 1937 to 1939 and studying under Pia Nalli. Then he went to the university of Rome, where in 1941 he earned his laurea with magna cum laude under the direction of Mauro Picone, when he was only 19. He was immediately appointed by Picone as an assistant professor to his chair and as a researcher at the Istituto Nazionale per le Applicazioni del Calcolo, becoming his pupil. After the war he went back to Rome working with Mauro Picone: in 1948 he became "Libero Docente" of mathematical analysis and in 1949 he was appointed as full professor at the University of Trieste. As he remembers in, in both cases one of the members of the judging commission was Renato Caccioppoli, which become a close friend of him. From 1956 onward he was full professor at the University of Rome in the chair of mathematical analysis and then at the Istituto Nazionale di Alta Matematica in the chair of higher analysis, succeeding to Luigi Fantappiè. He retired from university teaching in 1992, but was professionally very active until his death in 1996: particularly, as a member of the Accademia Nazionale dei Lincei and first director of the journal Rendiconti Lincei – Matematica e Applicazioni, he succeeded in reviving its reputation.

Honours

He was member of several academies, notably of the Accademia Nazionale dei Lincei, the Accademia Nazionale delle Scienze detta dei XL and of the Russian Academy of Science.

Teachers

His lifelong friendship with his teacher Mauro Picone is remembered by him in several occasions. As recalled by, his father Giuseppe was an assistant professor to the chair of Picone while he was teaching at the University of Catania: they become friends and their friendship lasted even when Giuseppe was forced to leave the academic career for economic reasons, being already the father of two sons, until Giuseppe's death. The young, in effect child, Gaetano, was kept by Picone in his arms. From 1939 to 1941 the young Fichera developed his research directly under the supervision of Picone: as he remembers, it was a time of intense work. But also, when he was back from the front in April 1945 he met Picone while he was in Roma in his way back to Sicily, and his advisor was so happy to see him as a father can be seeing its living child. Another mathematician Fichera was influenced by and acknowledged as one of his teachers and inspirators was Pia Nalli: she was an outstanding analyst, teaching for several years at the University of Catania, being his teacher of mathematical analysis from 1937 to 1939. Antonio Signorini and Francesco Severi were two of Fichera's teachers of the Roman period: the first one introduced him and inspired his research in the field of linear elasticity while the second inspired his research in the field he taught him i.e. the theory of analytic functions of several complex variables. Signorini had a strong long-time friendship with Picone: on a wall of the apartment building where they lived, in Via delle Tre Madonne, 18 in Rome, a memorial tablet which commemorates the two friends is placed, as recalls. The two great mathematicians extended their friendship to the young Fichera, and as a consequence this led to the solution of the Signorini problem and the foundation of the theory of variational inequalities. Fichera's relations with Severi were not as friendly as with Signorini and Picone: nevertheless, Severi, which was one of the most influential Italian mathematicians of the first half of the 20th century, esteemed the young mathematician. During a course on the theory of analytic functions of several complex variables taught at the Istituto Nazionale di Alta Matematica from the fall of 1956 and the beginning of the 1957, whose lectures were collected in the book, Severi posed the problem of generalizing his theorem on the Dirichlet problem for holomorphic function of several variables, as recalls: the result was the paper, which is a masterpiece, although not generally acknowledged for various reasons described by. 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 recalls.

Friends

A complete list of Fichera's friends includes some of the best scientists and mathematicians of the 20th century: Olga Oleinik, Olga Ladyzhenskaya, Israel Gel'fand, Ivan Petrovsky, Vladimir Maz'ya, Nikoloz Muskhelishvili, Ilia Vekua, Richard Courant, Fritz John, Kurt Friedrichs, Peter Lax, Louis Nirenberg, Ronald Rivlin, Hans Lewy, Clifford Truesdell, Edmund Hlawka, Ian Sneddon, Jean Leray, Alexander Weinstein, Alexander Ostrowski, Renato Caccioppoli, Solomon Mikhlin, Paul Naghdi, Marston Morse were among his friends, scientific collaborators and correspondents, just to name a few. He built up such a network of contacts being invited several times to lecture on his research by various universities and research institutions, and also participating to several academic conferences, always upon invitation. This long series of scientific journeys started in 1951, when he went to the USA together with his master and friend Mauro Picone and Bruno de Finetti in order to examine the capabilities and characteristics of the first electronic computers and purchase one for the 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 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 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 ?"–. Gaetano quickly answered:– "Non solo ti darò la condizione sufficiente, ma ti darò anche la condizione necessaria e pure per insiemi non-limitati "–. In effect, Fichera proved such a theorem in the paper, 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, as 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 : 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 elasticity theory includes the paper, where Fichera proves the "Fichera's maximum principle", his work on variational inequalities. The work on this last topic started with the paper, where he announced the existence and uniqueness theorem for the Signorini problem, and ended with the following one, where the full proof was published: those papers are the founding works of the field of variational inequalities, as remarked by Stuart Antman in. Concerning the Saint-Venant's principle, he was able to prove it using a variational approach and a slight variation of a technique employed by Richard Toupin to study the same problem: in the paper there is a complete proof of the principle under the hypothesis that the base of the cylinder is a set with piecewise smooth boundary. Also he is known for his researches in the theory of hereditary elasticity: the paper 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 and for Siegfried Flügge's Handbuch der Physik.

Partial differential equations

He was one of the pioneers in the development of the abstract approach through functional analysis in order to study general boundary value problems for linear partial differential equations proving in the paper a theorem similar in spirit to the 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,, he proves the first existence theorem for the mixed boundary problem for self-adjoint operators of variables, while in the paper he proves the same theorem dropping the hypothesis of self-adjointness. He is, according to, the founder of the theory of partial differential equations of non-positive characteristics: in the paper 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, which is written in English and was later translated in Russian and Hungarian.

Calculus of variation

His contributions to the calculus of variation are mainly devoted to the proof of existence and uniqueness theorems for maxima and minima of functionals of particular form, in conjunction with his studies on variational inequalities and linear elasticity in theoretical and applied problems: in the paper a semicontinuity theorem for a functional introduced in the same paper is proved in order to solve the Signorini problem, and this theorem was extended in to the case where the given functional has general linear operators as arguments, not necessarily partial differential operators.

Functional analysis and eigenvalue theory

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, where an important existence theorem is proved.
His contributions in the field of eigenvalue theory began with the paper, 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, this method does not give any estimate on the approximation error on the value of the calculated eigenvalues.
He contributed also to the classical eigenvalue problem for symmetric operators, introducing the method of orthogonal invariants.

Approximation theory

His work in this field is mainly related to the study of systems of functions, possibly being particular solutions of a given partial differential equation or system of such equations, in order to prove their completeness on the boundary of a given domain. The interest of this research is obvious: given such a system of functions, every solution of a boundary value problem can be approximated by an infinite series or Fourier type integral in the topology of a given function space. One of the most famous examples of this kind of theorem is Mergelyan's theorem, which completely solves the problem in the class of holomorphic functions for a compact set in the complex plane. In his paper, Fichera studies this problem for harmonic functions, relaxing the smoothness requirements on the boundary in the already cited work : a survey on his and others' work in this area, including contributions of Mauro Picone, Bernard Malgrange, Felix Browder and a number of other mathematicians, is contained in the paper. Another branch of his studies on approximation theory is strictly tied to complex analysis in one variable, and to the already cited Mergelyan's theorem: he studied the problem of approximating continuous functions on a compact set of the complex plane by rational functions with prescribed poles, simple or not. The paper 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 potential theory are very important. The results of his paper occupy paragraph 24 of chapter II of the textbook, as remarked by in. Also, his researches and on the asymptotic behaviour of the electric field near singular points of the conducting surface, widely known among the specialists can be included in between his works in potential theory.

Measure and integration theory

His main contributions to those topics and are the papers and. In the first one he proves that a condition on a sequence of integrable functions previously introduced by Mauro Picone is both necessary and sufficient in order to assure that limit process and the integration process commute, both in bounded and unbounded domains: the theorem is similar in spirit to the dominated convergence theorem, which however only states a sufficient condition. The second paper contains an extension of the Lebesgue's decomposition theorem to finitely additive measures: this extension required him to generalize the Radon–Nikodym derivative, requiring it to be a set function belonging to a given class and minimizing a particular functional.

Complex analysis of functions of one and several variables

He contributed to both the classical topic of complex analysis in one variable and the more recent one of complex analysis in several variables. His contributions to complex analysis in one variable are essentially approximation results, well described in the survey paper. In the field of functions of several complex variables, his contributions were outstanding, but also not generally acknowledged. Precisely, in the paper he solved the Dirichlet problem for holomorphic function of several variables under the hypothesis that the boundary of the domain has a Hölder continuous normal vector and the Dirichlet boundary condition is a function belonging to the Sobolev space satisfying the weak form of the tangential Cauchy–Riemann condition, extending a previous result of Francesco Severi: this theorem and the Lewy–Kneser theorem on the local Cauchy problem for holomorphic functions of several variables, laid the foundations of the theory of CR-functions. Another important result is his proof in of an extension of Morera's theorem to functions of several complex variables, under the hypothesis that the given function is only locally integrable: previous proofs under more restrictive assumptions were given by Francesco Severi in and Salomon Bochner in. He also studied the properties of the real part and imaginary part of functions of several complex variables, i.e. pluriharmonic functions: starting from the paper 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, and generalizes a theorem of Luigi Amoroso to the complex vector space for complex variables in the paper. Also he was able to prove that an integro-differential equation defined on the boundary of a 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.

Exterior differential forms

His contributions to the theory of exterior differential forms started as a war story: having read a famous memoir of Enrico Betti just before joining the army, he used this knowledge in order to develop a theory of exterior differential forms while he was kept prisoner in Teramo jail. When he was back in Rome in 1945, he discussed his discovery with Enzo Martinelli, who very tactfully informed him that the idea was already developed by mathematicians Élie Cartan and Georges de Rham. However, he continued work on this theory, contributing with several papers, and also advised all of his students to study it, despite from the fact of being an analyst, as he remarks: his main results are collected in the papers and. In the first one he introduced -measures, a concept less general than currents but easier to work with: his aim was to clarify the analytic structure of currents and to prove all relevant results of the theory i.e. the three theorems of de Rham and Hodge theorem on harmonic forms in a simpler, more analytic way. In the second one he developed an abstract Hodge theory, following the axiomatic method, proving an abstract form of Hodge theorem.

Numerical analysis

As noted in the "Functional analysis and eigenvalue theory" section, his main direct contribution to the field of numerical analysis is the introduction of the method of orthogonal invariants for the calculus of eigenvalues of symmetric operators: however, as already remarked, it is hard to find something in his works which is not related to applications. His works on partial differential equations and linear elasticity have always a constructive aim: for example, the results of paper, which deals with the asymptotic analysis of the potential, were included in the book and led to the definition of the as a standard benchmark problem for numerical methods. Another example of his work on quantitative problems is the interdisciplinary study, surveyed in, where methods of mathematical analysis and numerical analysis are applied to a problem posed by biological sciences.

History of mathematics

his work in this field occupy all the volume. 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. 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" and in the volume. 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