Steady PDEs on generalized rectangles: a change of genericity in mode interactions

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Abstract
Boundary conditions may induce subtle effects on the genericity of bifurcation problems. The group of transformations that leave the domain invariant does not necessarily contain all the symmetries of the problem. We give a detailed description of how this phenomenon occurs for elliptic PDEs (in particular steady reaction-diffusion equations) with Neumann boundary conditions on n-dimensional rectangles. The generic bifurcation equations for mode interactions are given in the appropriate symmetry context. We observe that the form of the equations does not depend on the dimension of the domain. With a suitable interpretation and some extra minor considerations, the classification of Armbruster and Dangelmayr (1987) for the 1-dimensional case applies in n dimensions as well.