# Summer Graduate School

Parent Program: |
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Location: |
Japan - Okinawa Institute of Science and Technology |

## Show List of Lecturers

- Ugur Abdulla (Okinawa Institute of Science and Technology)
- Gui-Qiang Chen (University of Oxford)
- Alessia Kogoj (Università di Urbino)
- Monica Torres (Purdue University)

## Show List of Teaching Assistants

- Federica Gregorio (Università di Salerno)
- Daniel Tietz (Okinawa Institute of Science and Technology)

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

**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.**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.

#### Prerequisites

- Basic Measue Theory, Distribution Theory, Sobolev Spaces, Functional Analysis
- 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.**

#### Venue

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

**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**

26B20 - Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)

35L50 - Initial-boundary value problems for first-order hyperbolic systems

35L67 - Shocks and singularities for hyperbolic equations [See also 58Kxx, 74J40, 76L05]

76L05 - Shock waves and blast waves in fluid mechanics [See also 35L67]

**Secondary Mathematics Subject Classification**

26B30 - Absolutely continuous real functions of several variables, functions of bounded variation

28A25 - Integration with respect to measures and other set functions

28A75 - Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]

74J40 - Shocks and related discontinuities in solid mechanics