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## Abstract / Description of output

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional
J(u):=∫Ω|∇u|p+λp+χ{u>0}+λp−χ{u≤0},1<p<∞.
Here $\Omega \subset \R^N$ is a bounded smooth domain and λ± are positive constants such that λp+−λp−>0. We prove the following dichotomy: if x0 is a free boundary point then either the free boundary is smooth near x0 or u has linear growth at x0. Furthermore, we show that for p>1 the free boundary has locally finite perimeter and the set of non-smooth points of free boundary is of zero (N−1)-dimensional Hausdorff measure. Our approach is new even for the classical case p=2.

Original language | English |
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Pages (from-to) | 40-81 |

Number of pages | 33 |

Journal | Advances in Mathematics |

Volume | 328 |

Early online date | 12 Jan 2018 |

DOIs | |

Publication status | Published - 13 Apr 2018 |

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