Geometric function theory, inverse problems and fluid dinamics
Principal investigator: Daniel Faraco
Project reference: 307179
Start Date: 2012-10-01
End Date: 2017-09-30
Abstract: The project will strike for conquering frontier results in three capital areas in partial differential equations and mathematical analysis: Elliptic equations and systems, fluid dynamics and inverse problems.
I propose to tackle the central problems in these areas with a new perspective based on the theory of differential inclusions. A thorough study of oscillating div-curl couples in this framework will lead us to the long expected higher dimensional version of the Tartar conjecture. The corresponding analysis of differential inclusions for gradient fields will lead to new results respect to the existence, uniqueness and regularity theory on the so far intractable theory of higher dimensional Beltrami systems.
Next we will concentrate in weak solutions to the classical nonlinear equations governing fluid dynamics. A reformulation of these equations as differential inclusions enables a much more rich theory of weak solutions than the classical one. With this new tool at hand, we will close several long standing questions about existence, uniqueness and contour dynamics.
The third part of the project is devoted to inverse problems in p.d.e. The most famous inverse problem is Calderon conductivity problem which asks whether the Dirichlet to Neumann map of an elliptic equation determines the coefficients. The problem is still open in three or more dimensions but a new formulation as a differential inclusion will allow us to close the 1980 Calderon conjecture by constructing new invisible materials. In dimension n=2 the recent approach based on quasiconformal theory will lead to the first regularization scheme valid for discontinuous conductivities and first results for nonlinear equations. For the stationary Schrodinger equation I propose to exploit a fascinating connection with the convergence to initial data of the non-elliptic time dependent Schrodinger equation.
"This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 307179"