Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidiication. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this...

Consider the equation −ε 2 ∆u ε + q(x)u ε = f (u ε) in R 3 , |u(∞)| < ∞, ε = const > 0. Under what assumptions on q(x) and f (u) can one prove that the solution u ε exists and lim ε→0 u ε = u(x), where u(x) solves the limiting problem q(x)u = f (u)? These are the questions discussed in the paper.