Nonexistence of higher dimensional stable turing patterns in the singular limit

When the thickness of the interface (denoted by ε) tends to zero, any stable stationary internal layered solutions to a class of reaction-diffus on systems cannot have a smooth limiting interfacial configuration. This means that if the limiting configuration of the interface has a smooth limit, it must become unstable for small ε, which makes a, sharp contrast with the one-dimensional case. This suggests that stable layered patterns must become very fine and complicated in this singular limit. In fact we can formally derive that the rate of s irinking of stable patterns is of order ε^1/3. Using this scaling, the resulting rescaled reduced equation determines the morphology of magnified patterns. A variational characterization of the critical eigenvalue combined with the matched asymptotic expansion method is a key ingredient for the proof, although the original linearized system is not of self-adjoint type.