# Analysis and Partial Differential Equations

## 12th – 16th June 2023  Okinawa, Japan

Summer graduate school will provide five days of concentrated study of topics in Analysis and Partial Differential Equations. The school will offer two minicourses and several plenary lectures on the topics at the forefront of current research in Partial Differential Equations and Potential Theory. The school will be well suited for math graduate students and advanced undergraduate math majors with a wide range of abilities and knowledge. The structure of the courses will be aimed to create an active learning environment through a combination of classical and flipped classroom teaching.

If you would like to attend this Summer Graduate School, please apply here by the 5th of February 2023.

All inquiries should be directed to miwako.tokuda@oist.jp.

Organizing committee:

Lecturers:

Plenary Lecturers:

• Prof. Sir John Ball, FRS, FRSE, Sedleian Professor of Natural Philosophy, University of Oxford, Heriot-Watt University, United Kingdom
• Prof. Gui-Qiang G. Chen, Statutory Professor in the Analysis of PDEs, Director of the Oxford Centre for Nonlinear PDEs (OxPDE), University of Oxford, United Kingdom
• Prof. John Lewis, University of Kentucky, USA
• Prof. Juan Manfredi, University of Pittsburgh, USA
• Prof. Ugur G. Abdulla, Professor and Head of the Analysis & PDE Unit, Okinawa Institute of Science and Technology (OIST), Okinawa, Japan

Teaching Assistants:

• Dr. Yanyan Guo, Postdoctoral Scientist, Analysis & PDE Unit, OIST
• Dr. Daniel Tietz, Postdoctoral Scientist, Analysis & PDE Unit, OIST
• Dr. Firoj Sk, Postdoctoral Scientist, Analysis & PDE Unit, OIST

Scientific Description: SGS will offer two mini-courses during the one-week-long school:

Course I: Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs (Ugur Abdulla & Alessia Kogoj)

Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results in XX-century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. In this course, we will present some recent developments precisely characterizing the regularity of the point at $$\infty$$ for second-order elliptic and parabolic PDEs and broadly extending the role of the Wiener test in classical analysis. We preface the description of the course with a citation from a classical paper by Wiener: the Dirichlet Problem (DP) divides itself into two parts, the first of which is the determination of the harmonic function corresponding to certain boundary conditions, while the second is the investigation of the behavior of this function in the neighborhood of the boundary. In the first part of the course, we focus on proving the existence of the solution to the DP for the Laplace equation, and its parabolic counterpart for the heat/diffusion equation. Solvability, in some generalized sense, of the Dirichlet problem in an arbitrary open set with prescribed data on its topological boundary is realized within the class of resolutive boundary functions, identified by Perron's method, and its Wiener and Brelot refinements. Such a method is referred to as the PWB method and the corresponding solutions are the PWB solutions. In the second part of the course, our focus will be on the boundary regularity of PWB solutions. The regularity of a boundary point is a problem of local nature and it depends on the measure-geometric properties of the boundary in the neighborhood of the boundary point and the differential operator. After introducing the concept of Newtonian capacity we discuss Wiener’s celebrated criterion which expresses the boundary regularity in terms of the divergence of the Wiener integral with the integrand being a weighted capacity of the exterior set in the neighborhood of the boundary point. The high point of the course is the new concept of regularity or irregularity of the point at $$\infty$$ defined according as the harmonic measure of $$\infty$$ is null or positive, and the discussion of the proof of the new Wiener criterion for the regularity of $$\infty$$. The Wiener test at $$\infty$$ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at $$\infty$$ arises as a thinness criterion at $$\infty$$ in fine topology. In a probabilistic context, the Wiener test at $$\infty$$ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criterion for the removability of singularities and/or for unique continuation at the finite boundary point.

References

• Course II:   Hamilton-Jacobi equations in metric spaces (Qing Liu & Xiaodan Zhou)

In this course, we discuss the solvability and various properties of first-order Hamilton-Jacobi equations in metric space. The Hamilton-Jacobi equations in the Euclidean spaces are widely applied in various fields such as optimal control, geometric optics, computer vision, and image processing. It is also well known that the notion of viscosity solutions provides a successful framework for the well-posedness of first order fully nonlinear equations. Recent years have witnessed a growing interest and significant progress in extending the viscosity solution theory to general metric spaces, motivated by rapid developments of optimal transport, control theory, traffic flow, data sciences etc. The purpose of the course is to introduce a few key achievements of this topic, including the definitions of metric viscosity solutions, comparison theorems and existence results, and provide an overview of new developments on related problems and applications. It is worth remarking that several different notions of metric viscosity solutions have been proposed for the eikonal equations, a special class of Hamilton-Jacobi equations. We will elaborate on this aspect, showing the equivalence of these notions under appropriate assumptions on the metric spaces and the structure of the equations. As a related application, the eigenvalue problem associated to the infinity Laplacian will also be discussed.  In order to help students better understand the materials, at the beginning of the course we will provide sufficient background and go over some preliminaries about metric spaces and Hamilton-Jacobi equations in Euclidean spaces.

References

• L. Ambrosio, J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces, J. Differential Equations, 256 (7) (2014) 2194–2245.
• M. Bardi, I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1997.
• D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.
• W. Gangbo, A. Święch, Optimal transport and large number of particles, Discrete Contin. Dyn. Syst. 34 (4) (2014) 1397–1441.
• W. Gangbo, A. Święch, Metric viscosity solutions of Hamilton-Jacobi equations depending on local slopes, Calc. Var. Partial Differ. Equ. 54 (1) (2015) 1183–1218.
• Y. Giga, N. Hamamuki, A. Nakayasu, Eikonal equations in metric spaces, Trans. Am. Math. Soc. 367 (1) (2015) 49–66.
• C. Imbert, R. Monneau, H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var. 19 (1) (2013) 129–166.
• Q. Liu, A. Mitsuishi, Principal eigenvalue problem for infinity Laplacian in metric spaces, to appear in Adv. Nonlinear Stud.
• Q. Liu, A. Nakayasu, Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces, Discrete Contin. Dyn. Syst. 39 (1) (2019) 157–183.
• Q. Liu, N. Shanmugalingam, and X. Zhou, Equivalence of solutions of eikonal equation in metric spaces, J. Differential Equations 272 (2021), 979–1014.
• A. Nakayasu, Metric viscosity solutions for Hamilton-Jacobi equations of evolution type, Adv. Math. Sci. Appl. 24 (2) (2014) 333–351.
• A. Nakayasu, T. Namba, Stability properties and large time behavior of viscosity solutions of Hamilton-Jacobi equations on metric spaces, Nonlinearity 31 (11) (2018) 5147–5161

• Participants: School will be well suited for an audience of graduate students with a wide range of abilities and knowledge. Both courses are designed in a way that the prerequisite material mostly overlaps with the material taught in the standard graduate courses on Analysis and PDEs. Therefore, even first-year graduate students with solid undergraduate math backgrounds will have the potential for a comfortable and engaging start. The structure of the courses will aim to create an active learning environment through a combination of classical and flipped classroom teaching. Every working day each course will present a classical-style lecture given by the professor followed by an active learning session. The goal of the second session is twofold: first to sharpen the comprehension of the material of the given lecture, and second, to prepare students for the forthcoming lecture. Both professors and assistants will be involved in active learning sessions to help students. Starting from day 2, active learning sessions will include student presentations of the assignments given in the previous day’s lecture. To address the variance of academic backgrounds, students will be divided into groups, each group including students with varying backgrounds. Each group will have at least one presentation during the course, which will include participation of all group members. In order to make active learning sessions more effective an online discussion forum will be created, and students will be encouraged to post their questions and comments following every lecture. This discussion forum will define the major discussion topics of the following active learning sessions.

Lesson plan/Syllabus: Daily schedule of the SGS on Mo-Fr (except W) will be   as follows:

• 9:00-10:15. Course I – Lecture 1.
• 10:30-11:45. Course I – Lecture 2/active learning session
• 12:00-13:00. Lunch
• 13:30-14:30. Plenary Lecture
• 14:30-15:00. Tea and Coffee
• 15:00-16:15. Course II – Lecture 1.
• 16:30-17:45. Course II – Lecture 2/active learning session

Wednesday's schedule is as follows:

•  9:00-10:15. Course I – Lecture 1.
• 10:30-11:45. Course II – Lecture
• 12:00-13:00. Lunch
• 13:30-18:00. Field trip
• 19:00-21:00. Conference Banquette

Course I:  Syllabus

1. Perron’s method for the Laplacian (Alessia Kogoj)

Harmonic functions, Gauss mean value formula, superharmonic functions, solution of the Dirichlet problem for Euclidean balls, the Poisson kernel, generalized solution of the Dirichlet problem in the sense of Perron-Wiener-Brelot, boundary behavior of the PWB solution, Bouligand's theorem.

2. Wiener criterion at $$\infty$$ for the elliptic PDEs (Ugur Abdulla)

Newtonian potentials, Newtonian capacity, minimization of energy functionals in Hilbert space setting, Wiener criterion at finite boundary points, geometric tests for boundary regularity, harmonic measure, regularity of $$\infty$$, Wiener criterion for the regularity of  $$\infty$$, fine topology, asymptotic laws for the Brownian motion.

Course II:  Syllabus

1. Preliminaries on metric spaces and Hamilton-Jacobi equations (Xiaodan Zhou)

Metric spaces, rectifiable curves, geodesic spaces, length spaces, induced intrinsic metric, slopes of functions, Hamilton-Jacobi equations in Euclidean spaces, eikonal equations, viscosity solutions, comparison principle, Perron’s method, optimal control interpretation

2. Metric viscosity solutions and applications (Qing Liu)

Hamilton-Jacobi equations in metric spaces, metric viscosity solutions, comparison principle, Ekeland’s variational principle, equivalent definitions of solutions to eikonal equations, infinity-eigenvalue problem in metric spaces, comparison with cones

Prerequisites:

In the Graduate Textbook: Lawrence C. Evans, Partial Differential Equations, AMS, 2nd edition, 2010:

• Reviewing calculus facts outlined in Appendix C: Calculus
• Reviewing facts outlined in Appendices D and E: Foundational Analysis and Measure Theory
• Review Section 2.2. Laplace’s Equation; and Section 2.3. Heat Equation;
• Solve exercises 2-17 from Section 2.3 Problems.
• Review Section 2.4, Section 3, and Section 5

Location description: The description of the facilities for the SGS is given in the following link