Existence of weak solutions to volume-preserving mean curvature flow with obstacles

We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.