Hokkaido University Preprint Series in Mathematics
We are concerned with variational properties of a fold energy for a unit, dilation-invariant gradient field (called a cluster) in the unit disk. We show that boundedness of a fold energy implies L1-compactness of clusters. We also show that a fold energy is L1-lower semicontinuous. We characterize absolute minimizers. We also give a sequence of stationary states and discuss its stability. Surprisingly, the stability depends upon q, the power of modulus of the jump discontinuities, in the definition of the fold energy.