A free boundary problem with a volume penalization

LEDERMAN, CLAUDIA

ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA CL. DI SCIENZE - IV

Scuola Normale Superiore

Lugar: Pisa; Año: 1996 vol. 23 p. 249 - 300

0391-173X

In this paper we study the problem of minimizing                      Je (u) := òΩ((1/2)| su|2 – 2u) dx + fe(|{x є Ω/u(x) > 0}|),     u є K c, where,                                       fe ( s) :=      e( s – w0)                                   s≤ w0 ,    fe ( s) :=      (1/e)(( s – w0) + (s- w0)2)          s≥ w0 (|S| denotes the Lebesgue measure of a set S). The class Kc consists of all functions v in H1(Ω)∩ L1(Ω) such that v=c on ∂Ω. Here Ω is an unbounded domain in Rn, more precisely Ω:= Rn H,  where H is a bounded domain;  c, ε, w0 are positive constants. The present variational problem is motivated by the following optimal design problem: “Among all cylindrical elastic bars, with cross-section of a given area and with a single given hole in it, find the one with the maximum torsional rigidity”. This application is studied in a forthcoming paper where we solve this problem for holes belonging to a certain class. Given a hole in that class, we prove --denoting by w0 the area of the cross-section and by H the hole-- that the optimal cross-section is given by the set {u>0}, where u is the solution to the present variational problem for an appropriate pair of constants c and ε. The aim of this paper is to prove that a solution to our minimization problem exists and to study regularity properties of any solution u and the corresponding free boundary Ω∩∂{u>0}: We show that any solution is Lipschitz continuous and that the free boundary is locally analytic except --possibly-- for a closed set of (n-1)-dimensional Hausdorff measure zero. Moreover, in two dimensions singularities cannot occur, i.e. the free boundary is locally analytic. In addition, we show that, for small values of the penalization parameter ε, the volume of {u>0} automatically adjusts to w0.