Grant Details:

2018-2022: Excellence Hubs

Regularity in Free Boundary Problems

Acronym: RegFBPs

Grant Agreement Number: EXCELLENCE/1216/0025

Budget: 160,344 EUR

Funded by: European Regional Development Fund and the Republic of Cyprus through the Research and Innovation Foundation

Webpage: http://www.mas.ucy.ac.cy/~emilakis/REGinFBPs/index.html

The proposed project is included in the general area of linear and fully nonlinear differential equations and the theory of Free Boundaries. Partial Differential equations are perhaps the most important link between mathematics and other sciences. Models that appear in Physics, Biology, in Finance etc., are described by means of partial differential equations and the mathematical reasoning is essential for understanding and solving the corresponding problems. The main purpose of this project is to develop the mathematical methodology which will be suitable for a rigorous mathematics analysis of questions included in the area of Free Boundary Problems, and particularly for obstacle type problems. Several free boundary problems arise naturally while studying physical phenomena. These theoretical problems are motivated by applications in elasticity, in phase change of materials, flows of liquids, cavitation, flame propagation and questions in the gen- eral field of shape optimization. The area due to the nature of the problems (direct relationship with technology, natural and economical sciences) remains of topical interest. Obstacle problems are characterized by the fact that the solution must satisfy unilateral constraint i.e. must remain, on its domain of definition or part of it, above a given function the so called obstacle. Parabolic obstacle problems, i.e. when the involved operators are of parabolic type, can be formulated in various ways such as a system of inequalities, variational inequalities, Hamilton- Jacobi equation, etc. Recently, there is an intense interest, perhaps due to the connectivity to jump or anomalous diffusion, to study obstacle type problems when the operator involved is a non-local operator and especially the fractional Laplacian. The proposed research will produce lasting results and the primary theory for problems with immediate connections to applications that involve linear and nonlinear elliptic and parabolic equations as well as nonlocal operators for obstacle type and related free boundary problems.

The main objective of the project is the investment in research excellence. As such, the main objective of the Programme ”Excellence Hubs” will be fulfilled through the successful development of novel technics in topics that are presently considered to be at the forefront of academic basic research in the internationally competitive area of Partial Differential Equations with Free Boundaries.