@article {
author = {Aghajani, A.},
title = {A two-phase free boundary problem for a semilinear elliptic equation},
journal = {Bulletin of the Iranian Mathematical Society},
volume = {40},
number = {5},
pages = {1067-1086},
year = {2014},
publisher = {Iranian Mathematical Society (IMS)},
issn = {1017-060X},
eissn = {1735-8515},
doi = {},
abstract = {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.},
keywords = {Free boundary problems,optimal growth,regularity,singular set},
url = {http://bims.iranjournals.ir/article_553.html},
eprint = {http://bims.iranjournals.ir/article_553_0532226b244f965cdb1172d8caf97706.pdf}
}