Perturbation and Duality of Boundary Value Problems for Elliptic and Parabolic PDEs

The solvability theory for complex valued elliptic PDEs and systems is significantly less understood than the equivalent scalar elliptic theory with real coefficients. This is due to the lack of certain tools available in the scalar real case such as the
maximum principle or the De Giorgi-Nash-Moser regularity theory. Recently as new concept called p-ellipticity has allowed to make significant progress in the setting of elliptic complex valued PDEs. The aim of the project is to further explore this breakthrough and its consequences to solvability of such PDEs under the Regularity/Neumann boundary conditions and questions of extrapolation of solvability. The second topic the project will look at is p-ellipticity for elliptic systems with
particular focus on specific systems such as the Lame equations for linear elasticity.