2000/07/10 by Peter Kuchment, Yehuda Pinchover
Mathematics · #math.AP #math.SP #msc:35B05 #msc:35C15 #msc:58J15 #msc:35J15 #msc:35P05 #msc:58J50
published as J. Funct. Anal. 181 (2001), 402--446. · 48 pages
arxiv created 2000/07/10 · arxiv updated 2009/11/30
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.