### Abstract

We consider minimizers of F=(λ1(Ω),..λN(Ω))+ |Ω| where F is a function nondecreasing in each parameter, and λ_{k}(Ω) is the k^{th} Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often-studied F=λ_{N}. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler-Lagrange equation for the minimizers that is not otherwise available.

Original language | English (US) |
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Pages (from-to) | 1678-1721 |

Number of pages | 44 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 72 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2019 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*72*(8), 1678-1721. https://doi.org/10.1002/cpa.21810