Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media

Matteo Cozzi, Enrico Valdinoci

Research output: Contribution to journalArticle

Abstract

We consider here a nonlocal phase transition energy in a periodic medium and we construct solutions whose interfaces lie at a bounded distance from any given hyperplane. These solutions are either periodic or quasiperiodic, depending on the rational dependency of the normal direction to the reference hyperplane. Remarkably, the oscillations of the interfaces with respect to the reference hyperplane are bounded by a universal constant times the periodicity scale of the medium. This geometric property allows us to establish, in the limit, the existence of planelike nonlocal minimal surfaces in a periodic structure. The proofs rely on new optimal density and energy estimates. In particular, roughly speaking, the energy of phase transition minimizers is controlled, both from above and below, by the energy of one-dimensional transition layers.

Original languageEnglish
Article number3013
JournalNonlinearity
Volume31
Issue number7
DOIs
Publication statusPublished - 25 May 2018

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Periodic Media
Ginzburg-Landau
Perimeter
hyperplanes
Minimizer
Hyperplane
Fractional
Phase transitions
Periodic structures
Phase Transition
Energy
Transition Layer
Density Estimates
Energy Estimates
Periodic Structures
Minimal surface
minimal surfaces
Time Constant
Periodicity
transition layers

Cite this

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Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media. / Cozzi, Matteo; Valdinoci, Enrico.

In: Nonlinearity, Vol. 31, No. 7, 3013, 25.05.2018.

Research output: Contribution to journalArticle

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