Analysis of Elliptic and Parabolic Partial Differential Equations Based on Carleson Measures

Abstract

This thesis proposes a new technique to analyze nonlinear elliptic and parabolic partial differential equations by exploiting the knowledge generated by the associated Carleson measure characterization of admissible data classes. As an illustration, we study three selected problems: <br /> 1. The solvability and regularity questions pertaining to the inhomogeneous Dirichlet problem for the weakly harmonic map equations in various geometries. This constitutes the first part of the thesis. <br /> 2. In the second part, two strongly coupled systems of equations modelling chemotaxis driven processes of cells in presence of an incompressible fluid are analyzed. Local and global well-posedness results as well as uniqueness criteria are obtained. <br /> 3. The third part is devoted to the solvability and regularity problems for the stationary Navier-Stokes system subject to a non-smooth Dirichlet data.