Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X40520141001A two-phase free boundary problem for a semilinear elliptic equation10671086553ENA.AghajaniIran University of Science and Technology0000-0002-6656-113XJournal Article20130214In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $D\subset \mathbb{R}^{n}$ with smooth boundary. We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose Laplacians enjoy a certain inequality. We show that in dimension $n=2$, solutions have optimal growth at non-isolated singular points, and the same result holds for $n\geq3$ under an ($n-1$)-dimensional density condition. Furthermore, we prove that the set of points in the singular set that the solution does not have optimal growth is locally countably ($n-2$)-rectifiable.http://bims.iranjournals.ir/article_553_0532226b244f965cdb1172d8caf97706.pdf