Appl Math Optim 47:45–58 (2003)
2002 Springer-Verlag New York Inc.
A Property of Sobolev Spaces and Existence in Optimal Design
Institute of Mathematics, Romanian Academy,
P.O. Box 1-764, RO-70700 Bucharest, Romania
Abstract. We prove that for bounded open sets with continuous boundary,
Sobolev spaces of type W
() are characterized by the zero extension outside of
. Combining this with a compactness result for domains of class C, we obtain a
general existence theorem for shape optimization problems governed by nonlinear
nonhomogenous Dirichlet boundary value problems of arbitrary order, in arbitrary
dimension and with general cost functionals.
Key Words. Open sets of class C, Stability of Sobolev spaces, Compactness.
AMS Classiﬁcation. 49D37, 65K10.
The literature concerning existence theory for shape optimization problems is very rich.
There are several types of results: using regularity assumptions for the boundary of
the unknown domains (see  and ), using certain capacitary constraints , ,
 or using the notion of a generalized perimeter and constraints or penalty terms
constructed with it , . In the second case, conditions on the dimension of the
underlying Euclidean space have to be imposed in order to obtain the compactness of
certain families of open sets with respect to the Hausdorff–Pompeiu distance.
In this work, in arbitrary dimension, we study bounded open sets of class C, in the
sense of Maz’ya  or, equivalently, with the segment property, according to Adams .
In Section 2 we prove a compactness result in this class of open sets. An announce-
ment of this result may be found in , in a different context.
Section 3 proves a property for Sobolev spaces which may be compared with the
classical result: if z ∈ H
(D) and z = 0 quasi-everywhere in D − (where ⊂ D are