Planar loops with prescribed curvature: Existence, multiplicity and uniqueness results

Abstract

Let <!-- MATH $k:\mathbb{C}\to \mathbb{R}$ --> be a smooth given function. A -loop is a closed curve in <!-- MATH $\mathbb{C}$ --> having prescribed curvature at every point . We use variational methods to provide sufficient conditions for the existence of -loops. Then we show that a breaking symmetry phenomenon may produce multiple -loops, in particular when is radially symmetric and somewhere increasing. If 0$"> is radially symmetric and non-increasing, we prove that any embedded -loop is a circle; that is, round circles are the only convex loops in <!-- MATH $\mathbb{C}$ --> whose curvature is a non-increasing function of the Euclidean distance from a fixed point. The result is sharp, as there exist radially increasing curvatures 0$"> which have embedded -loops that are not circles.