Integral representations and Liouville theorems for solutions of periodic elliptic equations
Abstract
The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such solutions. The Liouville type theorems previously proved by M. Avellaneda and F.H. Lin, and J. Moser and M. Struwe for periodic second order elliptic equations in divergence form are significantly extended. Relations of these theorems with the analytic structure of the Fermi and Bloch surfaces are explained.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2000
 DOI:
 10.48550/arXiv.math/0007051
 arXiv:
 arXiv:math/0007051
 Bibcode:
 2000math......7051K
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Spectral Theory;
 Primary 35B05;
 35C15;
 58J15;
 Secondary 35J15;
 35P05;
 58J50
 EPrint:
 48 pages