We study the one and $N$-dimensional optimal profiles for the interface in a nonlocal model for phase transitions described by the free energy $$ F(u) := {1\over 4} \int\int J(x'-x) ( u(x') - u(x) )^2 dx' dx + \int W(u(x)) dx \ , $$ where $u$ is a scalar density function, $W$ is a double-well potential, and $J$ is a positive anisotropic interaction potential. We prove under very general assumptions that these profiles are one-dimensional.
A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem
ALBERTI, GIOVANNI;
1998-01-01
Abstract
We study the one and $N$-dimensional optimal profiles for the interface in a nonlocal model for phase transitions described by the free energy $$ F(u) := {1\over 4} \int\int J(x'-x) ( u(x') - u(x) )^2 dx' dx + \int W(u(x)) dx \ , $$ where $u$ is a scalar density function, $W$ is a double-well potential, and $J$ is a positive anisotropic interaction potential. We prove under very general assumptions that these profiles are one-dimensional.File in questo prodotto:
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