History Signorini problem

the classical signorini problem: equilibrium configuration of orange spherically shaped elastic body resting on blue rigid frictionless plane?

the problem posed antonio signorini during course taught @ istituto nazionale di alta matematica in 1959, later published article (signorini 1959), expanding previous short exposition gave in note published in 1933. signorini (1959, p. 128) himself called problem ambiguous boundary conditions, since there 2 alternative sets of boundary conditions solution must satisfy on given contact point. statement of problem involves not equalities inequalities, , not priori known of 2 sets of boundary conditions satisfied @ each point. signorini asked determine if problem well-posed or not in physical sense, i.e. if solution exists , unique or not: explicitly invited young analysts study problem.

gaetano fichera , mauro picone attended course, , fichera started investigate problem: since found no references similar problems in theory of boundary value problems, decided attack starting first principles, precisely virtual work principle.

during fichera s researches on problem, signorini began suffer serious health problems: nevertheless, desired know answer question before death. picone, being tied strong friendship signorini, began chase fichera find solution: fichera himself, being tied signorini similar feelings, perceived last months of 1962 worrying days. finally, on first days of january 1963, fichera able give complete proof of existence , uniqueness of solution problem ambiguous boundary condition, called signorini problem honour teacher. preliminary research announcement, later published (fichera 1963), written , submitted signorini week before death: , satisfied see positive answer question.

a few days later, during conversation family doctor damiano aprile, signorini told him:

according antman (1983, p. 282) solution of signorini problem coincides birth of field of variational inequalities.