# Heat flow from polygons

## TL;DR

This paper analyzes the initial heat flow from a polygonal domain in the plane with mixed boundary conditions, providing precise asymptotics for the heat content as time approaches zero.

## Contribution

It offers a detailed calculation of the heat content for polygons with mixed boundary conditions, including asymptotic behavior as time tends to zero.

## Key findings

- Derived asymptotic expansion of heat content as t→0
- Quantified the influence of boundary conditions on heat flow
- Provided explicit formulas for polygons with mixed boundary conditions

## Abstract

We study the heat flow from an open, bounded set $D$ in $\R^2$ with a polygonal boundary $\partial D$. The initial condition is the indicator function of $D$. A Dirichlet $0$ boundary condition has been imposed on some but not all of the edges of $\partial D$. We calculate the heat content of $D$ in $\R^2$ at $t$ up to an exponentially small remainder as $t\downarrow 0$.