Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces
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Calculus of Variations
Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces Angelo Profeta1 · Karl-Theodor Sturm1 Received: 21 June 2019 / Accepted: 30 April 2020 © The Author(s) 2020
Abstract We introduce the transportation-annihilation distance W p 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 doubling of the open set, the space obtained by gluing together two copies of it along the boundary. Mathematics Subject Classification 35K05 · 58J32 · 58J35 · 51F99 · 53C23 · 60B10 · 54E35 · 31E05
1 Introduction and statement of main results We present an approach to heat flow with homogeneous Dirichlet boundary conditions via optimal transport—indeed, the very first ever—based on a novel particle interpretation for this evolution. The classical particle interpretation for the heat flow in an open set Y with Dirichlet boundary condition is based on particles which move around in Y and are killed (or lose their mass) as soon as they hit the boundary ∂Y . Our new interpretation will be based on particles moving around in Y , which are reflected if they hit the boundary, and which thereby randomly change their “charge”: half of them change into “antiparticles”, half of them continue to be normal particles. Effectively, they annihilate each other but the total number of charged particles remains constant. This leads us to regard the initial probability distribution as a distribution σ0+ of normal particles, with no antiparticles being around at time 0, i.e. σ0− = 0. In the course of time, σt+
Communicated by A. Malchiodi.
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Karl-Theodor Sturm [email protected] Angelo Profeta [email protected]
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Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany 0123456789().: V,-vol
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and σt− will evolve as subprobability measures on Y and so does the “effective distribution” σt0 := σt+ −σt− whereas the “total distribution” σ t := σt+ +σt− continues to be a probability measure. The latter will evolve as heat flow with Neumann boundary conditions whereas the former will evolve as heat flow with Dirichlet boundary conditions. The evolution of the charged particle distribution σt = (σt+ , σt− ) will be characterized as an EVI-gradient flow for the Boltzmann entropy. New transportation distances for subprobability measures will yield contraction estimates for the effective flow. Technically, we will interpret the pairs of subprobability measures (σ + , σ − ) as a probability measure on the doubling of Y in X , i.e. a space obtained by gluing
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