It is well known that quasiconformal mappings govern planar linear elliptic equations and systems but they are also intimately linked with the theory of nonlinear elliptic systems of which not so much is known. Basic questions as uniqueness, regularity or topological properties of the solution depend on the particular properties of the nonlinearity. The theory of differential inclusions and subtle topological arguments provide new method to investigate the nonlinearities.
Our way to understand the higher dimension analogously quickly bifurcates into branches. On one hand the study of linear and nonlinear Beltrami systems is limited in the smooth cases by the Weyl and Cotton Yorks tensors and thus relates to conformal geometry. Still many of the planar ideas seem to be willing to evolve to the space and will be investigated. Again here the connections with the vectorial calculus of variation seems promising. On the other hand planar quasiconvexity and many properties of planar quasiconformal mappings seemed to be better reflected by the so-called Div-Curl couples introduced by Iwaniec and Sbordone. We expect that Div-Curl couples play the role on planar quasiconformal mappings in understanding elliptic equations. In particular the compactness properties might resemble the situation related to the Tartar conjecture proved by Faraco and Székelyhidi [Acta 2008].
Next we will concentrate in weak solutions to the classical nonlinear equations governing fluid dynamics. Last years have witnessed a number of breakthroughs in the area. On one hand reformulation of these equations as differential inclusions proposed by De Lellis and Székelyhidi [Annals 2009] enables a much more rich theory of weak solutions than the classical which allows for models which described classically ill posed problems. On the other hand, the theory of contour dynamics of the group of Diego Córdoba [Annals 2012] exhibit a zoo of interesting explicit weak solutions. An important goal of the project is relating these two issues.
The third part of the project is devoted to inverse problems in partial differential equations and geometry. The most famous inverse problem is Calderón 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 and it is related to unique continuation. Unique continuation fails in higher dimensions but it seems to us that the examples need to be better understood. Dimension n = 2 which in principle is harder has experienced large developments firstly due to the use of quasiconformal techniques due to Astala and Päivärinta [Annals 2006] and secondly by the solution of Calderón problem for general potentials in the framework of Schrödinger equations. We will pursue these two approaches in particular properties of scattering transforms, regularization schemes and stability. A major open problem in the field is to be able to deal with truly nonlinear equations. We will investigate this issue in the framework of the p-laplacian. We will investigate this issue in the framework of the p-laplacian. In addition, recent breaktrough of Dos Santos Ferreira, Kenig, Salo, and Uhlmann has shown that a tight relation between conformal geometry and the solvability of Calderón problem.
Negative results for the Calderón problem have led to models for electromagnetic quantum and acoustic cloaking. We believe that the theory of differential inclusions and a deeper understanding of the relation between G-convergence and Dirichlet to Neumann maps might shed new light in these connections and will allow to create a different type of invisible materials, possibly easier to manufacture.