# FY2020 Annual Report

**Analysis on Metric Spaces Unit
Assistant Professor Xiaodan Zhou**

## Abstract

Our unit starts from July 2020 and our first postdoc Dr. Antoni Kijowski joined us in November. Most of our unit activities in FY2020 have been carried out remotely including hosting a zoom mini-symposium "Partial differential equations under various metrics" on December 8-11, 2020.

Our research in the past year can be briefly summarized into the following two aspects: the first part is focused on the study of solutions to the first order Hamilton-Jacobi equations and Sobolev/BV functions on metric measure spaces; the second part emphasizes on horizontal convexity in the Heisenberg group and applications to sub-elliptic equations.

## 1. Staff

- Dr. Antoni Kijowski, Postdoctoral Scholar
- Chiyo Eto, Administrative Assistant

## 2. Collaborations

### 2.1 Equivalence of solutions of eikonal equation in metric spaces

- Description: Equivalence of solutions of eikonal equation in metric spaces
- Type of collaboration: Joint research
- Researchers:
- Professor Qing Liu, Fukuoka University
- Professor Nageswari Shanmugalingam, University of Cincinnati

### 2.2 Functions of bounded variation on complete and connected one-dimensional metric spaces

- Description: Functions of bounded variation on complete and connected one-dimensional metric spaces
- Type of collaboration: Joint research
- Researchers:
- Professor Panu Lahti, Universit at Augsburg

### 2.3 Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

- Description: Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations
- Type of collaboration: Joint research
- Researchers:
- Professor Qing Liu, Fukuoka University

### 3. Activities and Findings

### 3.1 Equivalence of solutions of eikonal equation in metric spaces

In this project we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rectifiably connected metric spaces. The notions considered are that of curve-based viscosity solutions, slope-based viscosity solutions, and Monge solutions. By using the induced intrinsic (path) metric, we reduce the metric space to a length space and show the equivalence of these solutions to the associated Dirichlet boundary problem. Without utilizing the boundary data, we also localize our argument and directly prove the equivalence for the definitions of solutions. Regularity of solutions related to the Euclidean semi-concavity is discussed as well.

### 3.2 Functions of bounded variation on complete and connected one-dimensional metric spaces

In this project, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda [18]. Furthermore, we study the necessity of conditions on the underlying space in Federer’s characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincar´e inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

### 3.3 Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

This project introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.

## 4. Publications

### 4.1 Journals

- Liu, Q., Shanmugalingam, N. and
**Zhou, X**.,*Equivalence of solutions of eikonal equation in metric spaces,*.**J. Differential Equations** -
Lahti, P. and

**Zhou, X**., Functions of bounded variation on complete and connected one-dimensional metric spaces, to appear in*International Mathematics Research Notices*. arXiv -
Liu, Q. and

**Zhou, X.**, Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations, to appear in*Ann. Sc. Norm. Super. Pisa Cl. Sci.*arXiv

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

- Zhou, X., Eikonal Equations on Metric Spaces, Partial Differential Equations under Various Metrics 2020, Okinawa, Japan, December 11 (2020)
- Zhou, X., Horizontal convex envelope of continuous functions in the Heisenberg group, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, September 23, 2020

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

**Mini-symposium Partial differential equations under various metrics, December 8-11**

### 6.1 Total variation minimization on graphs for semisupervised and unsupervised machine learning

- Date: December 8, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Andrea Bertozzi (University of California, Los Angeles)

### 6.2 Convexity preserving properties for Hamilton–Jacobi equations in geodesic spaces

- Date: December 8, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Atsushi Nakayasu (Kyoto University)

### 6.3 Motion of graph-like curves by crystalline surface diffusion

- Date: December 8, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Mi-Ho Giga (The University of Tokyo)

### 6.4 First order Mean Field Games on networks

- Date: December 8, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Claudio Marchi (University of Padova)

### 6.5 A finer singular limit of a single-well Modica-Mortola functional and its applications

- Date: December 9, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Yoshikazu Giga (University of Tokyo)

### 6.6 Certain min-max values related to the p-energy and packing radii

- Date: December 9, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Ayato Mitsuishi (Fukuoka University)

### 6.7 Gradient estimates for heat kernel and harmonic functions and applications

- Date: December 9, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Renjin Jiang (Tianjin University)

### 6.8 A regularity theory for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

- Date: December 9, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Fabio Camilli (University of Rome)

### 6.9 Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures

- Date: December 10, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Andrzej Swiech (Georgia Institute of Technology)

### 6.10 Elliptic and parabolic boundary value problems on rotationally symmetric domains

- Date: December 10, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Asuka Takatsu (Tokyo Metropolitan University)

### 6.11 Well-posedness and regularity for an H1-projection problem.

- Date: December 11, 2020
- Venue: OIST Campus Lab 4 and online
- Speaker: Wilfrid Gangbo (UCLA)

## 7. Other

Nothing to report.