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Summer Graduate School

Analysis of Partial Differential Equations (Okinawa Institute of Science and Technology) July 29, 2024 - August 09, 2024
Parent Program: --
Location: Japan - Okinawa Institute of Science and Technology
Organizers Ugur Abdulla (Okinawa Institute of Science and Technology), Gui-Qiang Chen (University of Oxford)
Lecturer(s)

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Teaching Assistants(s)

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Speaker(s)

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Description

This two week summer school, jointly organized by SLMath with OIST, will offer the following two mini-courses:

  1. Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form
    This course will present some recent developments in the theory of divergence-measure fields via measure-theoretic analysis and its applications to the analysis of nonlinear PDEs of conservative form – nonlinear conservation laws.
  2. Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs
    This course will present some recent developments precisely characterizing the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extending the role of the Wiener test in classical analysis.

School Structure

Each day will consist of two lectures and two problem sessions; one on each of the above courses.

Suggested Reading List

Prerequisites

  1. Basic Measure Theory, Distribution Theory, Sobolev Spaces, Functional Analysis
  2. In the graduate textbook: Lawrence C. Evans, Partial Differential Equations, 2nd edition, AMS, 2010:
  • Reviewing calculus facts outlined in Appendix C: Calculus
  • Reviewing facts outlined in Appendices D and E: Functional 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

Application Procedure

SLMath is only able to support a limited number of students to attend this school.  Therefore, it is likely that only one student per institution will be funded by SLMath.

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Venue

The summer school will be held at the Okinawa Institute of Science and Technology, Seaside House

OIST Webpage

https://groups.oist.jp/analysis-and-pde-2024/post/2023/10/17/summer-graduate-school-analysis-and-pde-okinawa


Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • potential theory

  • elliptic and parabolic PDEs

  • Laplace equation

  • heat equation

  • Dirichlet problem

  • super- and subharmonic functions

  • Wiener criterion

  • boundary regularity

  • regularity (or irregularity) of ∞

  • caloric function

  • super- and subcaloric functions

  • harmonic measure

  • parabolic measure

  • capacity

  • Newtonian potential

  • thermal capacity

  • thermal potential

  • Radon measure

  • fine topology

  • Brownian motion

  • Wiener processes

  • divergence-measure fields

  • PDE of divergence form

  • nonlinear conservation laws

  • hyerbolic conservation laws

  • sets of finite perimeter

  • BV functions

  • approximation

  • sets with Lipschitz boundary

  • open sets

  • Cauchy flux

  • balance laws

  • entropy solutions

  • foundation of continuum mechanics

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
Schedule, Notes/Handouts & Videos
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Jul 29, 2024
Monday
09:00 AM - 10:15 AM
  Course II – Lecture 1: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs"
Ugur Abdulla (Okinawa Institute of Science and Technology)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course II – Lecture 2/Active Learning Session (als): "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs"
Ugur Abdulla (Okinawa Institute of Science and Technology)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:15 PM
  Course II - Lecture 3: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs III"
Ugur Abdulla (Okinawa Institute of Science and Technology)
02:15 PM - 02:45 PM
  Afternoon Tea
02:45 PM - 04:00 PM
  Course II – Lecture 4/als: " Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs IV"
Ugur Abdulla (Okinawa Institute of Science and Technology)
Jul 30, 2024
Tuesday
09:00 AM - 10:15 AM
  Course I – Lecture 1: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form I"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course I – Lecture 2/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form II"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:15 PM
  Course I - Lecture 3: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form III"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
02:15 PM - 02:45 PM
  Afternoon Tea
02:45 PM - 04:00 PM
  Course I – Lecture 4/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form IV"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
Jul 31, 2024
Wednesday
09:00 AM - 10:15 AM
  Course II – Lecture 5: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs V"
Ugur Abdulla (Okinawa Institute of Science and Technology)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course II – Lecture 6/ als: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs VI"
Ugur Abdulla (Okinawa Institute of Science and Technology)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Plenary Lecture: TBA
Gui-Qiang Chen (University of Oxford)
02:00 PM - 02:30 PM
  Afternoon Tea
02:30 PM - 03:45 PM
  Course I - Lecture 5: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form V"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
03:45 PM - 05:00 PM
  Course I – Lecture 6/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form VI"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
Aug 01, 2024
Thursday
09:00 AM - 10:15 AM
  Course II – Lecture 7: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs VII"
Ugur Abdulla (Okinawa Institute of Science and Technology)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course II – Lecture 8/ als: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs VIII"
Ugur Abdulla (Okinawa Institute of Science and Technology)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:15 PM
  Course I - Lecture 7: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form VII"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
02:15 PM - 02:45 PM
  Afternoon Tea
02:45 PM - 04:00 PM
  Course I – Lecture 8/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form VIII"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
Aug 02, 2024
Friday
09:00 AM - 10:15 AM
  Course II – Lecture 9: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs IX"
Ugur Abdulla (Okinawa Institute of Science and Technology)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course II – Lecture 10/ als: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs "
Ugur Abdulla (Okinawa Institute of Science and Technology)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Plenary Lecture: TBA
Juan Manfredi
02:00 PM - 02:30 PM
  Afternoon Tea
02:30 PM - 04:45 PM
  Course I - Lecture 9: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form IX"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
03:45 PM - 05:00 PM
  Course I – Lecture 10/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form X"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
Aug 05, 2024
Monday
09:00 AM - 10:15 AM
  Course I - Lecture 11: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XI"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 11:45 AM
  Course I – Lecture 12/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XII"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Plenary Lecture: "Weak convergence methods for the nonlinear PDEs I"
Lawrence Evans (University of California, Berkeley)
02:00 PM - 02:30 PM
  Afternoon Tea
02:30 PM - 03:45 PM
  Course II – Lecture 11: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XI"
Ugur Abdulla (Okinawa Institute of Science and Technology)
03:45 PM - 05:00 PM
  Course II – Lecture 12/als: 'Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XII"
Ugur Abdulla (Okinawa Institute of Science and Technology)
Aug 06, 2024
Tuesday
09:00 AM - 10:15 AM
  Course I - Lecture 13: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XIII"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Lecture 14/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XIV"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Plenary Lecture: "Weak convergence methods for the nonlinear PDEs II"
Lawrence Evans (University of California, Berkeley)
02:00 PM - 02:30 PM
  Afternoon Tea
02:30 PM - 04:00 PM
  Course II – Lecture 13: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XIII"
Ugur Abdulla (Okinawa Institute of Science and Technology)
04:00 PM - 05:00 PM
  Course II – Lecture 14/als: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XIV"
Ugur Abdulla (Okinawa Institute of Science and Technology)
Aug 07, 2024
Wednesday
09:00 AM - 10:15 AM
  Course II - Lecture 15: "Peron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XV"
Ugur Abdulla (Okinawa Institute of Science and Technology)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course I – Lecture 15. Gui-Qiang Chen & Monica Torres, Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XV
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Plenary Lecture: "Kolmogorov Problem and Wiener-type criteria for the Removability of Fundamental Singularity for the Parabolic PDEs"
Ugur Abdulla (Okinawa Institute of Science and Technology)
02:00 PM - 02:30 PM
  Afternoon Tea
02:30 PM - 09:00 PM
  Field trip to Okinawa Aquarium
Aug 08, 2024
Thursday
09:00 AM - 10:15 AM
  Course I - Lecture 16: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XVI"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course I – Lecture 17/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XVII"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:15 PM
  Course II – Lecture 16: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XVI"
Ugur Abdulla (Okinawa Institute of Science and Technology)
02:15 PM - 02:45 PM
  Afternoon Tea
02:45 PM - 04:00 PM
  Course II – Lecture 17/als: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XVII"
Ugur Abdulla (Okinawa Institute of Science and Technology)
Aug 09, 2024
Friday
09:00 AM - 10:15 AM
  Course I - Lecture 18: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XVIII"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 12:00 PM
  Course I – Lecture 19/als: "Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form XIX"
Gui-Qiang Chen (University of Oxford), Monica Torres (Purdue University)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:15 PM
  Course II – Lecture 18: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XVIII"
Ugur Abdulla (Okinawa Institute of Science and Technology)
02:15 PM - 02:45 PM
  Afternoon Tea
02:45 PM - 04:00 PM
  Course II – Lecture 19/als: "Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs XIX"
Ugur Abdulla (Okinawa Institute of Science and Technology)