### Abstract

We continue the analysis of the two-phase free boundary problems initiated by ourselves, studying where we studied the linear growth of minimizers in a Bernoulli-type free boundary problem at the non-flat points and the related regularity of free boundary. There, among other things, we also defined the functional (Formula presented.) where x
_{0}
is a free boundary point, i.e. (Formula presented.) and u is a minimizer of the functional (Formula presented.) for some bounded smooth domain (Formula presented.) and positive constants (Formula presented.) with (Formula presented.). Here we show (Formula presented.) in discrete monotone at non-flat points x
_{0}
, when N = 2 and p is sufficiently close to 2, and then establish the linear growth of u. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

Original language | English |
---|---|

Pages (from-to) | 1073-1101 |

Number of pages | 29 |

Journal | Communications in Partial Differential Equations |

Volume | 43 |

Issue number | 7 |

DOIs | |

Publication status | Published - 3 Jul 2018 |

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*Communications in Partial Differential Equations*,

*43*(7), 1073-1101. https://doi.org/10.1080/03605302.2018.1499776

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*Communications in Partial Differential Equations*, vol. 43, no. 7, pp. 1073-1101. https://doi.org/10.1080/03605302.2018.1499776

**A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two.** / Dipierro, Serena; Karakhanyan, Aram L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

AU - Dipierro, Serena

AU - Karakhanyan, Aram L.

PY - 2018/7/3

Y1 - 2018/7/3

N2 - We continue the analysis of the two-phase free boundary problems initiated by ourselves, studying where we studied the linear growth of minimizers in a Bernoulli-type free boundary problem at the non-flat points and the related regularity of free boundary. There, among other things, we also defined the functional (Formula presented.) where x 0 is a free boundary point, i.e. (Formula presented.) and u is a minimizer of the functional (Formula presented.) for some bounded smooth domain (Formula presented.) and positive constants (Formula presented.) with (Formula presented.). Here we show (Formula presented.) in discrete monotone at non-flat points x 0 , when N = 2 and p is sufficiently close to 2, and then establish the linear growth of u. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

AB - We continue the analysis of the two-phase free boundary problems initiated by ourselves, studying where we studied the linear growth of minimizers in a Bernoulli-type free boundary problem at the non-flat points and the related regularity of free boundary. There, among other things, we also defined the functional (Formula presented.) where x 0 is a free boundary point, i.e. (Formula presented.) and u is a minimizer of the functional (Formula presented.) for some bounded smooth domain (Formula presented.) and positive constants (Formula presented.) with (Formula presented.). Here we show (Formula presented.) in discrete monotone at non-flat points x 0 , when N = 2 and p is sufficiently close to 2, and then establish the linear growth of u. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

KW - free boundary regularit

KW - monotonicity formula

KW - p-Laplace

KW - Two phase

UR - http://www.scopus.com/inward/record.url?scp=85061266207&partnerID=8YFLogxK

U2 - 10.1080/03605302.2018.1499776

DO - 10.1080/03605302.2018.1499776

M3 - Article

VL - 43

SP - 1073

EP - 1101

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 7

ER -