Journal of the
Korean Mathematical Society JKMS

We study a free boundary problem satisfying Bernoulli type boundary condition along which the gradient of a piecewise harmonic solution jumps zero to a given c0nstant value. In such problem, the free boundary splits the domain into two regions, the zero set and the harmonic region. Our main interest is to identify the global shape and the location of the zero set. In this paper, we find the lower and the upper bound of the zero set. In a convex domain, easier estimation of the upper bound and faster disk test technique are given to find a rough shape of the zero set. Also a simple proof on the convexity of zero set is given for a connected zero set in a convex domain.

Keywords: interface identification problem, maximum principle, energy functional minimization, harmonic potential theory.

MSC numbers: Primary 35R35; Secondary 31A25