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

Abstract:

Let $k:\mathbb {C}\to \mathbb {R}$ be a smooth given function. A $k$-loop is a closed curve $u$ in $\mathbb {C}$ having prescribed curvature $k(p)$ at every point $p\in u$. We use variational methods to provide sufficient conditions for the existence of $k$-loops. Then we show that a breaking symmetry phenomenon may produce multiple $k$-loops, in particular when $k$ is radially symmetric and somewhere increasing. If $k>0$ is radially symmetric and non-increasing, we prove that any embedded $k$-loop is a circle; that is, round circles are the only convex loops in $\mathbb {C}$ 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 $k>0$ which have embedded $k$-loops that are not circles.