# A two-phase free boundary problem for a semilinear elliptic equation

Document Type : Research Paper

Author

Iran University of Science and Technology

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.

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### History

• Receive Date: 14 February 2013
• Revise Date: 27 July 2013
• Accept Date: 27 July 2013
• First Publish Date: 01 October 2014