In 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.
Aghajani, A. (2014). A two-phase free boundary problem for a semilinear elliptic equation. Bulletin of the Iranian Mathematical Society, 40(5), 1067-1086.
MLA
A. Aghajani. "A two-phase free boundary problem for a semilinear elliptic equation". Bulletin of the Iranian Mathematical Society, 40, 5, 2014, 1067-1086.
HARVARD
Aghajani, A. (2014). 'A two-phase free boundary problem for a semilinear elliptic equation', Bulletin of the Iranian Mathematical Society, 40(5), pp. 1067-1086.
VANCOUVER
Aghajani, A. A two-phase free boundary problem for a semilinear elliptic equation. Bulletin of the Iranian Mathematical Society, 2014; 40(5): 1067-1086.