Singular and Degenerate Points for a Free Boundary Problem

As minimal surfaces, free boundaries are not always regular. Therefore the regularity theory for free boundaries naturally splits into two branches, finding criterias that implies regularity and classification of possible singular points.

This talk will focus on the second branch for a problem considered by R. Monneau and G.S. Weiss. They consider the problem $$\mathrm{\Delta }u=-{\chi }_{\}u>0\}},$$ ${\chi }_{\{u>0\}}$ is the characteristic function of the indicated set. They show that the free boundary is regular whenever the solution grows quadraticly away from the free boundary. This property is shown for some special kinds of solutions, for instance for variational solutions.

However they leave the question open wether this quadratic growth is true in general.

In this talk we will recapitualte the main results of Monneau's and Weiss' paper. We will also answer their open question and show the existence of non-regular free boundary points. The proof is based on Schauders' fixed point theorem and symetry. The existence of both kinds of non-regular free boundary points will be shown. The first kind is when the solution grows faster than quadraticly away from the free boundary, at these points the solution fails to have the optimal $C^{1,1}$-regularity and will be denoted singular points of the solution. The other kind is the degenerete points when the solution grows slower than quadratic.

This work is joint with G.S. Weiss