Heat Flow with Dirichlet Boundary Conditions via Optimal Transport and Gluing of Metric Measure Spaces

Understand

We introduce the transportation-annihilation distance $W_p^\sharp$ between subprobabilities and derive contraction estimates with respect to this distance for the heat flow with homogeneous Dirichlet boundary conditions on an open set in a metric measure space.

• We also deduce the Bochner inequality for the Dirichlet Laplacian as well as gradient estimates for the associated Dirichlet heat flow.
• For the Dirichlet heat flow, moreover, we establish a gradient flow interpretation within a suitable space of charged probabilities.
• In order to prove this, we will work with the \emph{doubling} of the open set, the space obtained by gluing together two copies of it along the boundary.