The lecture offers students an insight into selected techniques for the regularity of solutions to second-order elliptic equations. In the lecture students learn to analyze the fundamental solution, to apply Caccioppoli estimates and to work with Morrey and Campanato spaces. Furthermore, the Moser-iteration is covered by this lecture. The goal is the development of a toolbox of techniques for the regularity of solutions of elliptical equations.

Elliptical equations occur in various applications in physics, chemistry and biology. Examples are equations from electrostatics and gravitation. Furthermore, elliptical equations can be found as stationary problems describing the diffusion of a chemical in a solution. If one wants to calculate the solutions numerically in these cases, it turns out that the numerical methods usually converge faster the more regular the solution of the elliptical equation is. On the one hand, direct methods using the fundamental solution are treated. On the other hand, the H^m regularity and the Hölderegularity of solutions of elliptic equations are also proved.

Partial Differential Equations (Graduate Studies in Mathematics, Band 19) of Lawrence C. Evans (2010)
    An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs vof Mariano Giaquinta and Luca Martinazzi
    Elliptic Partial Differential Equations of Second Order of David Gilbarg and  Neil S. Trudinger (1977)