In this thesis, we will develop an efficient method to study a shape optimization problem involving principal eigenvalue of an elliptic operator. We will consider the problem of minimizing the principal eigenvalue of an elliptic operator with respect to the distribution of two conducting materials in arbitrary disjoint measurable subsets A and B, respectively, of a fixed domain O. It is known that there is an optimal arrangement of the conductivities to give the overall lowest conductivity for some particular geometries with Dirichlet boundary condition, thereby producing minimal heat flow. However, the actual optimal configuration may not be known. When the design region is a ball, it is conjectured [C. Conca, R. Mahadevan, and L. Sanz. In ESAIM: Proceedings, volume 27, pages 311-321. EDP Sciences, 2009] that the material with the highest conductivity will be concentrated in the center. But, it is shown in [C. Conca, A. Laurain, and R. Mahadevanm SIAM Journal on Applied Mathematics, 72(4):1238-1259, 2012] that even in this simple case, this is not true in general.;Since different conductivity will yield different spectral properties, one of our goals is to compute the principal eigenvalue of the elliptic eigenvalue problems for any given sigma(x) efficiently and accurately. This is the so-called forward problem in this study. We then look for the optimal distribution of conductivity which yields the minimal principal eigenvalue. This is the so-called optimization problem. Since the Dirichlet problem was discussed in [C. Conca, R. Mahadevan, and L. Sanz. In ESAIM: Proceedings, volume 27, pages 311-321. EDP Sciences, 2009] already, we focus on the Neumann problem. We propose a numerical approach based on the rearrangement method to find the optimal conductivity for one-dimensional interval and general domains in two dimensions.
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