PhD Program in Mathematics and Computer Science

During the period April 26 – May 16, 2023, Prof. Vasile Staicu (University of Aveiro) has given a PhD course while visiting the Department of Mathematics and Computer Sciences within the Visiting Professor Program of INdAM.

Title: Boundary value problems with nonsmooth and multivalued terms

Abstract:

The aim of this course is to present some methods and tools used in the study of nonlinear boundary value problems involving multivalued maps and nonsmooth functions and illustrate those methods with some recent results concerning existence and multiplicity of solutions.

We start by introducing some notions of multivalued analysis: multivalued maps and their continuity. Then we introduce some elements of nonsmooth analysis that we use to develop nonsmooth critical point theory succeeded in extending a big part of the smooth theory to nonsmooth functionals: the main tool in the variational method for solving boundary value problems, which consists of trying to find solutions for a given boundary value problem by looking for stationary points of a real functional defined on a space of functions in which the solution of the boundary value problem is assumed to lie. 

Degree theory is a basic tool of nonlinear analysis and produces powerful existence and multiplicity results for nonlinear boundary value problems. We present a generalization of Brouwer degree theory to multivalued perturbations of monotone type maps, developed in a joint paper with Aizicovici and Papageorgiou (Mem. Amer. Math. Soc., 196, 2008). Then we consider a nonlinear Dirichlet problem with nonsmooth potential (hemivariational inequality) and use variational method to prove two constant sign solutions (one positive and another negative) and then by using degree theoretical approach we prove the existence of a third one, nontrivial solution, distinct from the previous two.

We present several existence and multiplicity results, with sign information for the solutions of nonlinear boundary value problems driven by the Laplacian, the p-Laplacian and then try to extend such results to nonlinear boundary value problems driven by fractional order differential operators and to nonlocal pseudo-differential inclusions, possibly including obstacles or constraints. It is reasonable to expect that the pairing of nonlocal diffusion operators and set-valued reaction will provide a more realistic model for applications to such problems as quantum mechanics, image restoration, and financial mathematics, which typically present a high degree of uncertainty, rather than elliptic equations with smooth, single-valued reactions.

Lecture notes will be provided and open questions will be considered and analyzed with the Ph. D. students.

Schedule: 27/4 11-13 28/4 9-11 2/5 11-13 5/5 9-11 9/5 11-13 11/5 11-13 12/5 9-11 16/5 11-13 (room F)

Lecture Notes: Staicu Lecture Notes