## Abstract

We consider the Dirichlet problem for uniformly elliptic operators L = ∑a_{ij}D_{ij} with measurable coefficients a_{ij} in the unit ball B_{1} ⊂ R^{d}. A recent sensational result of Nikolai Nadirashvili states that there is no uniqueness of "weak" solutions to this problem if d ≥ 3. He constructed two sequences of linear elliptic operators with smooth coefficients {a^{0, k}_{ij}} and {a^{1, k}_{ij}}, which have the same ellipticity constant v > 0 and converge to the same functions a_{ij} almost everywhere (a.e.) in B_{1} as k → ∞, while the corresponding sequences of solutions {u^{o, k}} and {u^{1, k}} converge to two different functions; i.e., the Dirichlet problem has at least two "weak" solutions. In the present paper, we popularize and slightly generalize Nadirashvili's result: for an arbitrary constant ∧ > 0, we construct two sequences of linear elliptic operators with the same ellipticity constant v = v(∧) > 0 and the additional restriction \a^{0, k}_{ij} - a^{1, k}_{ij}\ ≤ ∧ for all i, j, k, which define two different "weak" solutions to the Dirichlet problem [N. S. Nadirashvili, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), pp. 537-550].

Original language | English (US) |
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Pages (from-to) | 879-895 |

Number of pages | 17 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1999 |

## Keywords

- Elliptic PDE
- Measurable coefficients
- Nonuniqueness