Zeitschrift f r Analysis und ihre Anwendungen Journal for Analysis and its Applications Volume 17 (1998), No. 4, 1021-1024 Domain Identification for a Nonlinear Elliptic Equation D. D. Trong Abstract. It is proposed to identify the domain ci C R’ of a nonlinear elliptic equation subject to given Cauchy data on part of the known outer boundary F and to the zero condition on the unknown inner boundary -y. It is proved that under the assumption ci = ci, the domain ci is uniquely determined. Keywords: Domain identification, nonlinear elliptic equations, zero Dirichiet condition AMS subject classification: 35R30, 35J60 Let ci C R’1 be a bounded domain limited by an outer boundary F and an inner boundary y, where F is known, but y is unknown. Let F: R" x R x R n x R 2 R be a continuously differentiable function. We consider the nonlinear partial differential equation F(x,u, Du, D 2 u)=O (zEci), (1) ( where u = u(x), Du = and D2u= 8u subject to the boundary conditions ulro = f, au = g, U1, = 0 (2) where F 0 is an open subset of F. In the present paper, we consider domains ci C R’ 1 satisfying ci=ci (3) where A is the interior of the set A. Our problem is to determine a pair (ci, u) satisfying (1) - ( 2). The case that u is a harmonic function and the interior boundary -y is a star- shaped Jordan curve was considered in [1]. The present paper extends [1] in two ways. First, our equation is a fully nonlinear elliptic one (satisfying the maximum principle) D. D. Trong: Hochiminh City Nat. Univ., Dept. Math. & Comp. Sci.,-227 Nguyen-Van Cu str., Q5 Hochiminh City, Vietnam ISSN 0232-2064 / $ 2.50 Heldermann Verlag Berlin