Solutions of Nonlinear Planar Elliptic Problems with Triangle Symmetry

✍ Scribed by Stanislaus Maier-Paape; Thomas Wanner

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 498 KB
• Volume: 136
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.3240

✦ Synopsis

In this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fife et al. (Physica D 100 (1997), 257 278). More precisely, we consider small amplitude solutions of 2u+*f (u)=0 in R 2 whose nodal domains consist of equilateral triangles tiling the plane. If this equation is suitably perturbed, then for generic f we prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up. Furthermore, we treat the non-generic rectangular cases which had to be excluded in Fife et al. (Physica D 100 (1997), 257 278) as well as other nodal domain structures.