Fattening and nonfattening phenomena for planar nonlocal curvature flows

Abstract

We discuss fattening phenomenon for the evolution of planar curves according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel $K$, which is symmetric, nonnegative, possibly singular at the origin, and satisfies appropriate integrability conditions. We prove a general result about uniqueness of the geometric evolutions starting from regular sets with positive $K$-curvature and we discuss the fattening phenomenon for the evolution starting from the cross, showing that this phenomenon is very sensitive to the strength of the interactions. As a matter of fact, we show that the fattening of the cross occurs for kernels with sufficiently large mass near the origin, while for kernels that are sufficiently weak near the origin such a fattening phenomenon does not occur. We also provide some further results in the case of the fractional mean curvature flow, showing that strictly starshaped sets have a unique geometric evolution. Moreover, we exhibit two illustrative examples of closed nonregular curves, the first with a Lipschitz-type singularity and the second with a cusp-type singularity, given by two tangent circles of equal radius, whose evolution develops fattening in the first case, and is uniquely defined in the second, thus remarking the high sensitivity of the fattening phenomenon in terms of the regularity of the initial datum. The latter example is in striking contrast to the classical case of the (local) curvature flow, where two tangent circles always develop fattening. As a byproduct of our analysis, we provide also a simple proof of the fact that the cross is not a $K$-minimal set for the nonlocal perimeter functional associated to $K$.