On the relaxed formulation of some shape optimization problems. (English) Zbl 0971.49024

Buttazzo, G.; Dal Maso, G.; Garroni, A.; Malusa, A.

Adv. Math. Sci. Appl. 7, No. 1, 1-24 (1997).

Summary: We study the relaxed formulation of the shape optimization problem with constraints \(\min_A\{\int_A j(x,u_A) d\lambda: A\) open \(\subset\Omega\), \(\lambda(A)\in T\), \(Lu_A= f\) in \(A\), \(u_A\in H^1_0(A)\}\), where \(\Omega\) is a bounded open set in \(\mathbb{R}^n\), \(n\geq 2\), \(j: \Omega\times\mathbb{R}\to\mathbb{R}\) is a Carathéodory function, \(\lambda\) is a nonnegative Radon measure on \(\Omega\) vanishing on all sets with harmonic capacity zero, \(T\) is a closed interval of \([0,\lambda(\Omega)]\), \(L\) is an elliptic operator on \(\Omega\), and \(f\in H^{-1}(\Omega)\).

MSC:

49Q10	Optimization of shapes other than minimal surfaces
49J45	Methods involving semicontinuity and convergence; relaxation

Keywords:

relaxation; shape optimization; Carathéodory function; Radon measure; elliptic operator