Long Time Behavior of a Two-Phase Optimal Design for the Heat Equation

摘要: We consider a two-phase isotropic optimal design problem within the context of transient heat equation. The objective is to minimize average dissipated thermal energy during fixed time interval $[0,T]$. time-independent material properties are taken as variables. A full relaxation for this was established in [A. Munch, P. Pedregal, and F. Periago, J. Math. Pures Appl. (9), 89 (2008), pp. 225-247] by using homogenization method. In paper, we study asymptotic behavior $T$ goes infinity solutions relaxed prove that they converge an corresponding optimization stationary Next necessary optimality conditions under equation use those characterize microstructure designs, which appears form sequential laminate rank at most $N$, spatial dimension. An analysis lets us that, large enough, order lamination is, fact, $N-1$. Several numerical experiments two dimensions complete our study.