Optimal control for the conformal Laplacian obstacle problem

TL;DR

This paper investigates an optimal control problem related to the conformal Laplacian obstacle problem on closed Riemannian manifolds, establishing conditions for optimal controls and states, and characterizing solutions on the standard sphere.

Contribution

It demonstrates that optimal controls coincide with states under positive Yamabe invariant and characterizes solutions on the sphere as standard bubbles.

Findings

Optimal controls equal their associated states when Yamabe invariant is positive.

Existence of smooth optimal controls inducing constant scalar curvature metrics.

On the sphere, standard bubbles are the unique optimal controls.

Abstract

We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal metric with constant scalar curvature. For the standard sphere, we prove that the standard bubbles -- namely conformal factor of metrics conformal to the standard one with constant positive scalar curvature -- are the only optimal controls and hence equal to their associated optimal state.