@article{THESIS,
      recid = {4837},
      author = {Morfe, Peter},
      title = {Homogenization of Interface Motions in the Parabolic  Scaling},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2022-08},
      number = {THESIS},
      pages = {272},
      abstract = {This thesis details recent results on the periodic  homogenization of interface motions in the parabolic  scaling regime.  

First, we treat phase transitions in  periodic media, developing a variational approach that  unifies the point of view of Barles and Souganidis with  Aubrey-Mather theory.  We show that (possibly non-smooth)  pulsating standing waves can be obtained as minimizers of a  Percival-type Lagrangian in the spirit of the latter  theory.  Pulsating standing waves are shown to generate the  recurrent plane-like minimizers of the energy and to  determine the macroscopic surface tension.  In the case of  laminar media, these functions are used to demonstrate a  number of pathologies, such as the non-differentiability of  the surface tension.  Additionally, we prove two  homogenization results in the sharp-interface limit, one  pertaining to laminar media and a restricted class of  initial data, and the other, to the Allen-Cahn equation  with a periodic dissipation term.  

Finally, we study a  class of non-variational curvature flows with periodic  coefficients.  By analyzing a degenerate elliptic equation  on the torus, we identify the macroscopic behavior of the  moving interface at points where its normal vector is  irrational.  To analyze the motion near rational normals,  we adapt an idea of Feldman and Kim from the study of  oscillating boundary value problems.  This leads to the  conclusion that in dimensions three and higher, the  macroscopic interface velocity is a discontinuous function  of the normal direction.  We prove that the associated  interface motion is well-posed in spite of the  discontinuities and conclude that homogenization occurs.},
      url = {http://knowledge.uchicago.edu/record/4837},
      doi = {https://doi.org/10.6082/uchicago.4837},
}