Appl Math Optim 54:205–221 (2006)
2006 Springer Science+Business Media, Inc.
Optimal Measures for Nonlinear Cost Functionals
Interdisciplinary Centre for Mathematical and Computational Modelling,
Warsaw University, Zwirki i Wigury 93, 02-089 Warsaw, Poland
Abstract. We study an optimization problem for an elliptic PDE where the un-
known is a measure. The state equation is a relaxed form of the classical Dirichlet
problem and the functional to minimize is nonlinear. We prove a regularity result for
the optimal measure, give a description of it and present some numerical examples.
Key Words. Gamma-convergence, Measure theory, Shape optimization.
AMS Classiﬁcation. 49Q10, 35J20, 35B20.
In this paper we study the problem of optimal measure for a relaxed elliptic equation.
Given a bounded domain in R
, and two functions f ∈ L
() and j: × R → R,
we consider the following optimization problem:
J (µ) =
j (x, u
) dx+ αµ()
() is the subset of Borel measures which are absolutely continuous with
respect to the capacity of sets, α is a nonnegative parameter and u
is the solution of the
−u + uµ = f in ,
u = 0on∂.
In the particular case j (x, s) = f (x)s, problem (1) is the minimization of the
so-called elastic compliance under a constraint of mass. This case has been studied by
This work is part of the European Research Training Network “Homogenization and Multiple Scales”
(HMS2000) under Contract HPRN-2000-00109.