Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries
Henrik Shahgholian, Georg Sebastian Weiss · International series of numerical mathematics · 2006
The first part of this paper is an announcement of a result to appear. We apply the Aleksandrov reflection to obtain regularity and stability of the free boundaries in the two-dimensional problem $$ \Delta u = \frac{{\lambda _ + }} {2}\chi \{ u > 0\} - \frac{{\lambda _ - }} {2}\chi \{ u 0 and γ− > 0. In the second part we show that the Kelvin reflection can be used in a similar way to obtain regularity of the classical obstacle problem $$ \Delta u = \chi \{ u > 0\}$$ in higher dimensions.