Appl Math Optim 49:183–199 (2004)
2004 Springer-Verlag New York Inc.
Shape Optimization for Semi-Linear Elliptic Equations
Based on an Embedding Domain Method
Institut f¨ur Mathematik MA 4-5, Technische Universit¨at Berlin,
10623 Berlin, Germany
Abstract. We study a class of shape optimization problems for semi-linear elliptic
equations with Dirichlet boundary conditions in smooth domains in R
. A part of the
boundary of the domain is variable as the graph of a smooth function. The problem
is equivalently reformulated on a ﬁxed domain. Continuity of the solution to the
state equation with respect to domain variations is shown. This is used to obtain
differentiability in the general case, and moreover a useful formula for the gradient
of the cost functional in the case where the principal part of the differential operator
is the Laplacian.
Key Words. Domain optimization, Semi-linear elliptic equations, Embedding
AMS Classiﬁcation. 49Q10, 49J50, 35J60.
Shape or domain optimization problems are optimization problems where the control
parameter is the geometry of the computational domain. In many cases the cost functional
depends in a highly non-linear way on the shape of the domain via the solution of a state
equation, e.g. a PDE. Such problems occur in many applications, starting from ﬂuid
mechanics (e.g. design of cars and airfoils) over mechanical structures up to medicine
(e.g. the detection of the shape of tumors).