### Contents

Subject 1:  Basics about Laplace equation and harmonic functions.

• Lecturer: Assoc. Prof. Trần Vĩnh Hưng
• References: Chapter 2, Partial Differential Equations, Lawrence Craig Evans.
• Abstract: I will cover some basics about Laplace equation and harmonic functions. I will also introduce the Perron method to obtain the existence of solutions to the Laplace equation.
• Content: Fundamental solutions; mean-value formulas; properties of harmonic functions; the Perron method.

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Subject 2: Topics on biharmonic equations.

• Lecturer: Assoc. Prof. Ngô Quốc Anh
• References: Polyharmonic boundary value problems của Gazzola, Grunau, và Sweers  (https://www.springer.com/gp/book/9783642122446).
• Abstract: This course aims to provide a quick introduction to biharmonic equations with applications either on bounded domains or on the whole space. In the first part of the course, several key ingredients including boundary conditions and fundamental solutions are presented. In the second part of the course, several prototype biharmonic equations with geometric background are discussed.
• Content: boundary conditions; fundamental solutions; mean-value formulas; poly subharmonic functions.

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Subject 3: Front propagation for reaction-diffusion equations in periodic media.

• Lecturer: Dr. Võ Hoàng Hưng.
• References: Ryzhik’s lectures

http://math.stanford.edu/~ryzhik/technion-lect14.pdf

http://math.stanford.edu/~ryzhik/toulouse-lect.pdf

• Abstract: In this series of lectures, I recall first some of the important tools in the theory of solutions of elliptic and parabolic partial differential equations such as the maximum principle and the Hopf lemma. We will then use them in combination with some of the spectral properties of the elliptic operators to study front propagation in the periodic domain described by the Fisher-KPP equation. Some important open problems will also be addressed to interested researchers.