### Abstract

We consider the quasiminima of the energy functional ∫ _{Ω} A(x, ∇ u) + F(x, u) dx, where A(x, ∇ u) ∼ |∇ u|^{p} and F is a double-well potential. We show that the Lipschitz quasiminima, which satisfy an equipartition of energy condition, possess density estimates of Caffarelli-Cordoba-type, that is, roughly speaking, the complement of their interfaces occupies a positive density portion of balls of large radii. From this, it follows that the level sets of the rescaled quasiminima approach locally uniformly hypersurfaces of quasiminimal perimeter. If the quasiminimum is also a solution of the associated PDE, the limit hypersurface is shown to have zero mean curvature and a quantitative viscosity bound on the mean curvature of the level sets is given. In such a case, some Harnack-type inequalities for level sets are obtained and then, if the limit surface if flat, so are the level sets of the solution.

Original language | English |
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Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 33 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Sep 2008 |

Externally published | Yes |

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*Calculus of Variations and Partial Differential Equations*, vol. 33, no. 1, pp. 1-35. https://doi.org/10.1007/s00526-007-0146-1

**Geometry of quasiminimal phase transitions.** / Farina, Alberto; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Geometry of quasiminimal phase transitions

AU - Farina, Alberto

AU - Valdinoci, Enrico

PY - 2008/9/1

Y1 - 2008/9/1

N2 - We consider the quasiminima of the energy functional ∫ Ω A(x, ∇ u) + F(x, u) dx, where A(x, ∇ u) ∼ |∇ u|p and F is a double-well potential. We show that the Lipschitz quasiminima, which satisfy an equipartition of energy condition, possess density estimates of Caffarelli-Cordoba-type, that is, roughly speaking, the complement of their interfaces occupies a positive density portion of balls of large radii. From this, it follows that the level sets of the rescaled quasiminima approach locally uniformly hypersurfaces of quasiminimal perimeter. If the quasiminimum is also a solution of the associated PDE, the limit hypersurface is shown to have zero mean curvature and a quantitative viscosity bound on the mean curvature of the level sets is given. In such a case, some Harnack-type inequalities for level sets are obtained and then, if the limit surface if flat, so are the level sets of the solution.

AB - We consider the quasiminima of the energy functional ∫ Ω A(x, ∇ u) + F(x, u) dx, where A(x, ∇ u) ∼ |∇ u|p and F is a double-well potential. We show that the Lipschitz quasiminima, which satisfy an equipartition of energy condition, possess density estimates of Caffarelli-Cordoba-type, that is, roughly speaking, the complement of their interfaces occupies a positive density portion of balls of large radii. From this, it follows that the level sets of the rescaled quasiminima approach locally uniformly hypersurfaces of quasiminimal perimeter. If the quasiminimum is also a solution of the associated PDE, the limit hypersurface is shown to have zero mean curvature and a quantitative viscosity bound on the mean curvature of the level sets is given. In such a case, some Harnack-type inequalities for level sets are obtained and then, if the limit surface if flat, so are the level sets of the solution.

UR - http://www.scopus.com/inward/record.url?scp=38049018991&partnerID=8YFLogxK

U2 - 10.1007/s00526-007-0146-1

DO - 10.1007/s00526-007-0146-1

M3 - Article

VL - 33

SP - 1

EP - 35

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 1

ER -