TY - JOUR
ID - 553
TI - A two-phase free boundary problem for a semilinear elliptic equation
JO - Bulletin of the Iranian Mathematical Society
JA - BIMS
LA - en
SN - 1017-060X
AU - Aghajani, A.
AD - Iran University of Science and Technology
Y1 - 2014
PY - 2014
VL - 40
IS - 5
SP - 1067
EP - 1086
KW - Free boundary problems
KW - optimal growth
KW - regularity
KW - singular set
DO -
N2 - 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.
UR - http://bims.iranjournals.ir/article_553.html
L1 - http://bims.iranjournals.ir/article_553_0532226b244f965cdb1172d8caf97706.pdf
ER -