# Local minimizers of functionals with multiple volume constraints

Oudet, Édouard; Rieger, Marc Oliver (2008). Local minimizers of functionals with multiple volume constraints. ESAIM: Control, Optimisation and Calculus of Variations, 14(4):780-794.

## Abstract

We study variational problems with volume constraints, i.e., with level sets of prescribed measure. We introduce a numerical method to approximate local minimizers and illustrate it with some two-dimensional examples. We demonstrate numerically nonexistence results which had been obtained analytically in previous work. Moreover, we show the existence of discontinuous dependence of global minimizers from the data by using a $\Gamma$-limit argument and illustrate this with numerical computations. Finally we construct explicitly local and global minimizers for problems with two volume constraints.

## Abstract

We study variational problems with volume constraints, i.e., with level sets of prescribed measure. We introduce a numerical method to approximate local minimizers and illustrate it with some two-dimensional examples. We demonstrate numerically nonexistence results which had been obtained analytically in previous work. Moreover, we show the existence of discontinuous dependence of global minimizers from the data by using a $\Gamma$-limit argument and illustrate this with numerical computations. Finally we construct explicitly local and global minimizers for problems with two volume constraints.

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