Institution: | a Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA b Institut für Mathematik, Universität Augsburg, D-86135, Augsburg, Germany |
Abstract: | In this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on
2 and study the geometry of their nodal domains. In particular, we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern, then the perturbed equation will have a unique nearby solution which is still doubly periodic, but for which the nodal line structure breaks up. Moreover, we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern, and we relate these solutions to the Cahn-Hilliard equation and spinodal decomposition. |