Analysis & PDE
Members
Prof. Diego Ribeiro Moreira
Prof. Cleon S. Barroso
Prof. Gleydson Chaves Ricarte
Prof. Raimundo Alves Leitão Júnior
Prof. José Ederson Melo Braga
Description
The study of Partial Differential Equations (PDE’s) started in the 18th century with the works of Euler, d’Alembert, Lagrange and Laplace. It appeared as a central tool in the description of Continuum Mechanics and more generally, as the main device in the analytical study of models in the physical sciences. The subject has close connections with problems coming from Physics, Engineering and other scientific disciplines. It has also been used in quite revolutionary ways to foster the development of many other branches of Mathematics.
The development of much of all our present day technology can be seen as a product of our science and can be directly linked to the leading and paramount role played by PDEs as a decisive ingredient in our contemporary research.
The research in Elliptic and Parabolic equations is very well developed in Mathematics and of great importance because its applications in a broader scope of other scientific disciplines such as fluid dynamics, phase transitions and mathematical finance. The past few decades have witnessed many new developments in the theory of fully nonlinear equations, homogenization issues and diffusive non local processes. These advancements have promoted a better understanding of problems in material science, pricing options in the stock market and combustion theory just to mention few examples.
Free boundary problems are also an interesting field related to PDEs. They are the central subject in the study of phenomena where phase transitions occur. They naturally arise when one attempts to describe a discontinuous change of behavior in a physical or biological quantity. Applications appear in stopping time for optimal control, ground water hydrology, plasticity theory, optimal design and problems in superconductivity. Typical examples are the evolution of an ice-water mixture, an elastic membrane constrained to stay within a given region and the description of laminar flames as an asymptotic limit for high energy activation.
The research interests of the PDE group at UFC lie in a confluence of areas among which we could mention: fully nonlinear, quasilinear, degenerate and singular Elliptic and Parabolic equations, calculus of variations, free boundary problems, potential theory and geometric measure theory. The research directions in the department include:
- Regularity theory of solutions;
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Geometry and regularity of their level surfaces;
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Asymptotic behavior of solutions;
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Singularities of solutions.
