Thomas Wanner
Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, Virginia 22030, USA

 

Solutions of nonlinear planar elliptic problems with triangle symmetry

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  1. Stanislaus Maier-Paape, Thomas Wanner:
    Solutions of nonlinear planar elliptic problems with triangle symmetry
    Journal of Differential Equations 136(1), pp. 1-34, 1997.

Abstract

In this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fife, Kielhöfer, Maier-Paape, Wanner (1997). More precisely, we consider small amplitude solutions of Δu+λf(u)=0\Delta u + \lambda f(u) = 0 in R2\mathbb{R}^2 whose nodal domains consist of equilateral triangles tiling the plane. If this equation is suitably perturbed, then for generic ff 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, Kielhöfer, Maier-Paape, Wanner (1997), as well as other nodal domain structures.

The published version of the paper can be found at https://doi.org/10.1006/jdeq.1996.3240.

Bibtex

@article{maier:wanner:97a,
   author = {Stanislaus Maier-Paape and Thomas Wanner},
   title = {Solutions of nonlinear planar elliptic problems with
            triangle symmetry},
   journal = {Journal of Differential Equations},
   year = 1997,
   volume = 136,
   number = 1,
   pages = {1--34},
   doi = {10.1006/jdeq.1996.3240}
   }