The Dirichlet Problem in the Light of Capacity Theory

Dirichlet problems are a class of PDE problems that arose in the early 19th century, the idea is to look for a function which solves a PDE in the interior region while the value of the function at the boundary of the region is given. The solvability of the Dirichlet problem is linked to the theory of capacities which was initiated by Gustave Choquet in 1950s. This theory introduced a mathematical analogue of a set’s ability to hold electrical charge with a given potential energy with respect to an idealised ‘flat’ ground. This project aims to describe the links between the Dirichlet problem, the theory of capacities and Wiener’s criterion, with the goal of extending the theory to a more general class of operators.

Liuhao Yu

The University of Western Australia

Liuhao is currently an UWA student having completed his undergraduate studies in mathematics and statistics. He plans to pursue a masters degree in mathematical or statistical fields. Liuhao is also interested in various fields including human biology, computer science and philosophy, especially logic. He is particularly focusing on harmonic analysis and capacity theory now.

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