Tonelli graduated from the University of Bologna in 1907; his Ph.D. thesis was written under the direction of Cesare Arzelà.^[2]

He is one of the founders of Modern Theory of Functions of Real Variables and his work on the Calculus of Variations is a milestone in analysis.^[3]

— Olga Arsenievna Oleinik, (Oleinik 1986, p. 28)

The present writer's father, W. H. Young, used to recall that this very question — what principle can we use as the foundation of the calculus of variations^[4] — had been put him by a young Italian mathematician. His reply was a question: "Can you use semicontinuity?" The young Italian was Leonida Tonelli. Semicontinuity was then still a recent concept, known only to a few. In the hands of Tonelli, it became an important tool in a fundamental new approach to the calculus of variations.

— Laurence Chisholm Young, (Young 1969, p. 153)

• Opere scelte, a cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle ricerche, 1900
• Fondamenti di Calcolo delle Variazioni. Zanichelli, Bologna, vol. 1: 1922,^[5] vol. 2: 1923
• Tonelli, Leonida (1925). "The Calculus of Variations". Bulletin of the American Mathematical Society. 31 (3–4): 163–172. doi:10.1090/s0002-9904-1925-04002-1. MR 1561014.
• Serie trigonometriche. Zanichelli, Bologna 1928^[6]

• Calculus of variations
• Fourier series
• Lebesgue integral
• Mathematical analysis

Biographical and general references edit

• Accademia Nazionale dei Lincei (2012), Annuario dell'Accademia Nazionale dei Lincei 2012 – CDX dalla Sua Fondazione (PDF) (in Italian), Roma: Accademia Nazionale dei Lincei, p. 734, archived from the original (PDF) on 2016-03-04, retrieved 2016-05-15. The "Yearbook" of the renowned Italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of information about its academic and scientific activities.
• Cesari, Lamberto (1946), "Leonida Tonelli (1885-1946)", Bollettino dell'Unione Matematica Italiana, Serie 3 (in Italian), 1 (1): 64–67, available from the Biblioteca Digitale Italiana di Matematica.
• Cesari, Lamberto (1986), "L'opera di Leonida Tonelli e la sua influenza nel pensiero scientifico del secolo", in Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; Cesari, L.; Ciliberto, C.; Cimmino, G.; Cinquini, S.; De Giorgi, E.; Faedo, S.; Fichera, G.; Galligani, I.; Ghizzetti, A.; Graffi, D.; Greco, D.; Grioli, G.; Magenes, E.; Martinelli, E.; Pettineo, B.; Scorza, G.; Vesentini, E. (eds.), Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei (in Italian), vol. 77, Roma: Accademia Nazionale dei Lincei, pp. 41–73, archived from the original on 2011-02-23, retrieved 2013-01-19. "The work of Leonida Tonelli and his influence on scientific thinking in this century" (English translation of the title) is an ample commemorative article, reporting recollections of the Author about teachers and colleagues, and a detailed survey of his and theirs scientific work, presented at the International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985).
• Cinquini, Silvio (1950), "Leonida Tonelli", Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie II (in Italian), 15 (1–4): 1–37, MR 1556828, Zbl 0036.00505.
• Faedo, Sandro (1986), "Leonida Tonelli e la scuola matematica pisana", in Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; Cesari, L.; Ciliberto, C.; Cimmino, G.; Cinquini, S.; De Giorgi, E.; Faedo, S.; Fichera, G.; Galligani, I.; Ghizzetti, A.; Graffi, D.; Greco, D.; Grioli, G.; Magenes, E.; Martinelli, E.; Pettineo, B.; Scorza, G.; Vesentini, E. (eds.), Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei (in Italian), vol. 77, Roma: Accademia Nazionale dei Lincei, pp. 89–109, archived from the original on 2011-02-23, retrieved 2013-01-19. "Leonida Tonelli and the Pisa mathematical school" is a survey of the work of Tonelli in Pisa and his influence on the development of the school, presented at the International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985). The Author was one of his pupils and, after his death, held his chair of mathematical analysis at the University of Pisa, becoming dean of the faculty of sciences and then rector: he exerted a strong positive influence on the development of the university.
• Fichera, Gaetano (1995), "Tre battaglie perdute da tre grandi matematici italiani", Atti del convegno di studi in memoria di Giuseppe Gemignani. Modena, 20 maggio 1994, Collana di Studi dell'Accademia (in Italian), vol. 11, Modena: Enrico Mucchi Editore on behalf of the Accademia Nazionale di Scienze, Lettere e Arti di Modena, pp. 9–28, MR 1385469 {{citation}}: External link in |publisher= (help). This paper, included in the Proceedings of the Study Meeting in Memory of Giuseppe Gemignani, is an account of the failures of Vito Volterra, Leonida Tonelli and Francesco Severi, when dealing with particular research problems during their career. An English translation of the title reads as:-"Three battles lost by three great Italian mathematicians".
• Oleinik, O. A. (1986), "Indirizzo di adesione", in Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; Cesari, L.; Ciliberto, C.; Cimmino, G.; Cinquini, S.; De Giorgi, E.; Faedo, S.; Fichera, G.; Galligani, I.; Ghizzetti, A.; Graffi, D.; Greco, D.; Grioli, G.; Magenes, E.; Martinelli, E.; Pettineo, B.; Scorza, G.; Vesentini, E. (eds.), Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei, vol. 77, Roma: Accademia Nazionale dei Lincei, p. 26, archived from the original on 2011-02-23, retrieved 2013-01-19. The brief "participating address" presented to the International congress on the occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985) by Olga Oleinik on behalf of the Moscow Mathematical Society.
• Rizza, Giovanni Battista (1986), "Indirizzo di adesione", in Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; Cesari, L.; Ciliberto, C.; Cimmino, G.; Cinquini, S.; De Giorgi, E.; Faedo, S.; Fichera, G.; Galligani, I.; Ghizzetti, A.; Graffi, D.; Greco, D.; Grioli, G.; Magenes, E.; Martinelli, E.; Pettineo, B.; Scorza, G.; Vesentini, E. (eds.), Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei, vol. 77, Roma: Accademia Nazionale dei Lincei, pp. 29–30, archived from the original on 2011-02-23, retrieved 2013-01-19. The brief "participating address" presented to the International congress on the occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985), by Giovanni Battista Rizza on behalf of the University of Parma: the scientific relations between Leonida Tonelli and the Department of Mathematics in Parma are described.
• Sansone, Giovanni (1946), "Leonida Tonelli", Annali di Matematica Pura ed Applicata, IV Serie (in Italian), XXV: XIII–XVI, doi:10.1007/BF02418076, Zbl 0061.00604. The commemoration of Tullio Levi Civita, Guido Fubini, Leonida Tonelli and Michele De Franchis published on the "Annali di Matematica". Note that the paper is not signed: the Author is disclosed by Cinquini (1950, p.1, footnote (1)).
• Sneddon, Ian (1986), "Indirizzo di adesione", in Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; Cesari, L.; Ciliberto, C.; Cimmino, G.; Cinquini, S.; De Giorgi, E.; Faedo, S.; Fichera, G.; Galligani, I.; Ghizzetti, A.; Graffi, D.; Greco, D.; Grioli, G.; Magenes, E.; Martinelli, E.; Pettineo, B.; Scorza, G.; Vesentini, E. (eds.), Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei, vol. 77, Roma: Accademia Nazionale dei Lincei, pp. 35–36, archived from the original on 2011-02-23, retrieved 2013-01-19. Some recollections about Mauro Picone and Leonida Tonelli, presented by Ian Sneddon as the "participating address" (Italian translation of the title of the contribution) to the International congress on the occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985) on behalf of the Royal Society of London and of the Royal Society of Edinburgh.

Scientific references edit

• Cafiero, Federico (1959), Misura e integrazione, Monografie matematiche del Consiglio Nazionale delle Ricerche (in Italian), vol. 5, Roma: Edizioni Cremonese, pp. VII+451, MR 0215954, Zbl 0171.01503. Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
• Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, NJ – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 981-238-043-4, MR 1962933, Zbl 1028.49001.
• Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; Cesari, L.; Ciliberto, C.; Cimmino, G.; Cinquini, S.; De Giorgi, E.; Faedo, S.; Fichera, G.; Galligani, I.; Ghizzetti, A.; Graffi, D.; Greco, D.; Grioli, G.; Magenes, E.; Martinelli, E.; Pettineo, B.; Scorza, G.; Vesentini, E., eds. (1986), Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei, vol. 77, Roma: Accademia Nazionale dei Lincei, p. 318, archived from the original on 2011-02-23, retrieved 2013-01-19. The proceedings of the International congress on the occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli, held in Rome on May 6–9, 1985, at the Accademia Nazionale dei Lincei.
• Young, L. C. (1969), Lectures on the Calculus of Variation and Optimal Control, Philadelphia–London–Toronto: W. B. Saunders, pp. xi+331, ISBN 9780721696409, MR 0259704, Zbl 0177.37801.

• O'Connor, John J.; Robertson, Edmund F., "Leonida Tonelli", MacTutor History of Mathematics Archive, University of St Andrews
•   Media related to Leonida Tonelli (mathematician) at Wikimedia Commons