Splintering and coarsening in a growth and inhibition system

A geometric variational problem is utilized to study pattern formation as an outcome of growth and inhibition in physical and biological systems. A perimeter term in the problem gives a growth force while a negative power of the Laplace operator, [Formula: see text], provides an inhibition force. Two phenomena, splintering and coarsening, which are sources of morphological instability, are investigated. Splintering, which means large pieces of structures break into small pieces, is favored by the inhibition force and opposed by the growth force. On the other hand coarsening, which means large pieces increase in size and small pieces shrink and vanish, is favored by the growth force and opposed by the inhibition force. Several thresholds in terms of the parameters of the problem for these phenomena are found.