Bifurcations on Hemispheres

ly, we may think of a differential operator ~ on M as a mapping on the smooth functions on M. That is, let C~(M) denote the space of smooth real-valued functions on M. Then 3' : C~(M) ~ C~(M). Bifurcations on Hemispheres 215 In our extension theory we consider only those operators that respect the symmetries on M and N. We define these operators as follows. There is a natural action of ISO(M) on C~(M) defined by u ~ g(u) where for u E C~(M) and g ~ ISO(M), g(u)(x) = u ( g l x ) . (5.1) The operator ~' is ISO(M)-invariant if for all u ~ C~(M) and g ~ G we have @(g(u)) = g(~'(u)). (5.2) The best-known example of an ISO(M)-invariant is the Laplace operator associated to the Riemannian structure on M, denoted by A. It follows easily that the semi-linear elliptic operator ~ defined by ~(u) = Au + f (u ) , (5.3) where f : N ~ N is smooth, is also ISO(M)-invariant. Of course, this operator is the steady-state reaction-diffusion operator that we have discussed previously. This example can be generalized as follows. Recall that Au = div grad(u), where grad(u) is the gradient vector field of u. More generally, we say that a second order differential operator on M is in divergence form if we can write ~(u) = divA + B, (5.4) where A is a vector field on M depending smoothly on x E M, u, and du (the differential of u) and B is a scalar function depending smoothly on x, u, and du. Definition 5.5. A differential operator ~ is said to be a second order quasilinear elliptic operator in divergence form if ~ is an elliptic operator in the form (5.4). Definition 5.6. Suppose that u is a solution of ~ on N. 1. We say that u satisfies Neumann boundary conditions (NBC) on N if for every ~E ~t and all x E ON n Fix(T), we have