Appl Math Optim (2013) 67:391–418
Linear Integro-differential Schrödinger and Plate
Problems Without Initial Conditions
Published online: 1 February 2013
© Springer Science+Business Media New York 2013
Abstract Via Carleman’s estimates we prove uniqueness and continuous depen-
dence results for the temporal traces of solutions to overdetermined linear ill-posed
problems related to Schrödinger and plate equation. The overdetermination is pre-
scribed in an open subset of the (space-time) lateral boundary.
Keywords Ill-posed problems · Linear integro-differential Schrödinger and plate
equations · Uniqueness · Continuous dependence results
Severely ill-posed problems for PDE’s are well-known and studied as for uniqueness
and continuous dependence on the data. Each mathematician working with PDE’s
perfectly knows that the Cauchy problem for elliptic equations, the spatial bound-
ary (not the initial-boundary) problem for hyperbolic equations and the backward
initial-boundary problem for parabolic equations are ill-posed, i.e. they contradict
Hadamard’s celebrated deﬁnition of a well-posed problem that greatly affected the
Mathematics of the ﬁrst half of the twentieth century.
On the contrary, in the second half of the last century a lot of interest, due to the
rushing on of Technology, was devoted to Inverse Problems, a branch of which con-
sists just of severely ill-posed problems, where severely means that no transformation
can be found in order to change such problems to well-posed ones, at least, say, when
working in classical or Sobolev function spaces of ﬁnite order.
Communicating Editor: Rogerto Triggiani.
The author is a member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica.
A. Lorenzi (
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via Saldini 50, 20133