Minimizing optimal transport for functions with fixed-size nodal sets

Abstract

Consider the class of zero-mean functions with fixed $L^{\infty }$ and $L^1$ norms and exactly $N\in \mathbb{N}$ nodal points. Which functions $f$ minimize $W_p(f_{+},f_{-})$, the Wasserstein distance between the measures whose densities are the positive and negative parts? We provide a complete solution to this minimization problem on the line and the circle, which provides sharp constants for previously proven ``uncertainty principle''-type inequalities, i.e., lower bounds on $N\cdot W_p(f_{+},f_{-})$. We further show that, while such inequalities hold in many metric measure spaces, they are no longer sharp when the non-branching assumption is violated; indeed, for metric star-graphs, the optimal lower bound on $W_p(f_{+},f_{-})$ is not inversely proportional to the size of the nodal set, $N$. Based on similar reductions, we make connections between the analogous problem of minimizing $W_p(f_{+},f_{-})$ for $f$ defined on $\Omega \subset {\mathbb{R}}^d$ with an equivalent optimal domain partition problem.