Signorini Problem - History

History

The problem was posed by Antonio Signorini during a course taught at the Istituto Nazionale di Alta Matematica in 1959, later published as the paper Signorini 1959, expanding a previous short exposition he gave in a note published in 1933. According to Signorini (1959, p. 128) himself, he called it problem with ambiguous boundary conditions since there are two alternative sets of boundary conditions the solution must satisfy on any given contact point, involving not only equalities but also inequalities, but it is not a priori known what of the two sets is satisfied for each point: he asked to determine if the problem is well-posed or not in a physical sense, i.e. if its solution exists and is unique or not, inviting young analysts to study the problem. Gaetano Fichera and Mauro Picone attended the course, and Fichera started to investigate the existence and uniqueness of the solutions: since there were no references to a similar problem in the theory of boundary value problems, he decided to study the problem starting from first principles, precisely from the virtual work principle. While the problem was under investigation, Signorini began to suffer serious health problems: nevertheless, he desired to know the answer to his question before his death. Picone, being tied by a strong friendship with Signorini, began to chase Fichera to find a solution, who, being himself tied to Signorini by similar feelings, perceived the last months of 1962 as worrying days. Finally, on the first days of January 1963, Fichera was able to give a complete proof of the existence and uniqueness of a solution for the problem with ambiguous boundary condition, which he called "Signorini problem" to honour his teacher. The preliminary note later published as Fichera 1963 was written up and submitted to Signorini exactly a week before his death: He was very satisfied to see a positive answer to his question. A few days later, he told his family Doctor Damiano Aprile:-"Il mio discepolo Fichera mi ha dato una grande soddisfazione (My disciple Fichera gave me a great contentment)."-"Ma Lei ne ha avute tante, Professore, durante la Sua vita (But you had many, Professor, during your life)"- replied Doctor Aprile, but also Signorini replied:-"Ma questa è la più grande (But this is the greatest one)"-. And those were his last words. According to Antman (1983, p. 282) the solution of the Signorini problem coincides with the birth of the field of variational inequalities.