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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)

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Teaching Assistants(s)

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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.


  1. Basic Measue Theory, Distribution Theory, Sobolev Spaces, Functional Analysis
  2. 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: Fundational 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.


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

Keywords and Mathematics Subject Classification (MSC)
  • 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