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.
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
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]
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