FBPSINSC · Analysis of Free Boundary Problems arising in science

Analysis of Free Boundary Problems arising in science

Several boundary value and 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 and questions in the general field of shape optimization. The proposed project is included in the general area of linear and fully nonlinear differential equations and Geometric Measure Theory. 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 proposed research will produce lasting results and the primary theory for problems with immediate connections to applications. The area due to the nature of the problems (direct relationship with technology, natural and economic sciences) remains of topical interest. The mathematical developments in the last two decades added new concepts such as notions of weak solutions, monotonicity formulas and blow up techniques. These new methods to linear and nonlinear free boundary problems have contributed to the production of mathematical results giving in that way a new perspective in the area. The main purpose of this project is to develop the mathematical methodology which will be suitable for a rigorous mathematics analysis of problems in the general areas of Partial Differential Equations and Geometric Measure Theory with a spesific focus on Stefan-type/Thick obstacle-type free boundary problems, lower dimensional/thin obstacle type problems, questions on oundary regularity in rough domains for elliptic measures and finally stability questions eigenvalues and eigenvectors in rough domains. In this framework, boundary regularity results for elliptic measures and the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are small perturbations of given operators in rough domains have been obtained. Also we studied the stability of solutions to nonlinear Neumann problems in Reifenberg flat domains as well as the stability of eigenvalues for the Neumann and Dirichlet Laplacian in very rough domains. Finally, we obtained optimal regularity results for the solution in thick, thin and penalized thick obstacle problems for differential equations of parabolic type. A high level international workshop, entitled “Analysis of PDEs: Theory, Methods and Applications”, was organized in Cyprus during the summer of 2014. It took place in the town of Protaras during the period Jun 29th-July 4th, 2014. The main purpose of the conference was to bring together leading experts and young researchers to discuss recent developments in Analysis, with emphasis on both theory and applications. More information can be found at the workshop’s webpage: http://www.mas.ucy.ac.cy/~emilakis/Workshop2014/AnalysisPDES.htm The proposed project has greatly contributed in academic interactions at the University of Cyprus. On a regular base, Milakis and his visitors delivered lectures and seminar talks for post-docs and advanced graduate students. In addition, Milakis offered several graduate PDEs and Analysis courses aimed for masters and doctoral students. As a result graduate students and young researchers have been attracted to work in the area. A website with all up to date information and resources has been created and maintained at the servers located in the University of Cyprus: http://www.mas.ucy.ac.cy/~emilakis/FBPsinSC/index.html

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