We consider the obstacle problem:
a^{ij}D_{ij}w = Chi_{w > 0} in B_1
with
w = psi on partial B_1
where we assume that the coefficients $a^{ij}$ belong to VMO, that the functions $w, psi geq 0$ belong to the Sobolev space $W^{2,p},$ and that $w$ satisfies the PDE pointwise almost everywhere. We show existence, uniqueness, regularity, and nondegeneracy of the solutions. These results allow us to begin the study of the regularity of the free boundary. In particular, we establish a measure theoretic version of the Caffarelli Alternative after showing a measure stability result for the contact sets. (This is a joint work with Ivan Blank.)
Speaker: Kubrom Teka, Kansas State University