| Abstract: |
| An optimization problem related to the $p$-Laplacian operator is investigated. The aim is to determine a density function from a rearrangement class generated by a step function such that the principal eigenvalue is as small as possible. This optimization problem can be recasted into a shape optimization problem of finding a set of fixed measure such that the first eigenvalue will be minimal. At first some qualitative aspects of the minimizer set are obtained. Then, nearly optimal sets which are approximations of the minimizer for specific ranges of the parameter values are given. For those values of parameters, we show that these nearly optimal sets are in good agreement with the minimizers. In order to derive the optimal shape, a numerical algorithm is proposed and it is proved that the numerical procedure converges to a local optimizer. To demonstrate the efficiency and robustness of the algorithm, several numerical examples are provided. This is a joint work with Farid Bozorgnia. |
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