### Abstract

If A and B are two bounded domains in ℝ_{n} and λ(A), λ(B) are the lowest eigenvalues of -Δ with Dirichlet boundary conditions then there is some translate, B_{x}, of B such that λ(A∩B_{x})<λ(A)+λ(B). A similar inequality holds for {Mathematical expression}.There are two corollaries of this theorem: (i) A lower bound for sup_{x} {volume (A∩B_{x})} in terms of λ(A), when B is a ball; (ii) A compactness lemma for certain sequences in W^{1, p}(ℝ_{n}).

Original language | English (US) |
---|---|

Pages (from-to) | 441-448 |

Number of pages | 8 |

Journal | Inventiones Mathematicae |

Volume | 74 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 1983 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'On the lowest eigenvalue of the Laplacian for the intersection of two domains'. Together they form a unique fingerprint.

## Cite this

Lieb, E. (1983). On the lowest eigenvalue of the Laplacian for the intersection of two domains.

*Inventiones Mathematicae*,*74*(3), 441-448. https://doi.org/10.1007/BF01394245