Reference for a Heat Process in a Wedge

Question

I would like to ask about an explicit suggestion/reference for the following type of heat processes:

Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a tip at 0, having some hyperplanes (passing through 0) as sides and, say, the piece of the unit cyllinder/sphere as a bottom. In particular, in the case of $\mathbb{R}^2$ this "wedge" is a sector, i.e. the domain bounded by two rays, shot from 0, and the sphere.

To consider a simple special case, consider the sectorial domain $W = \{ (r, \theta) : r \geq 0, \theta \in [0, \alpha]\}$ in $\mathbb{R}^2$.Consider the following Dirichlet heat equation: \begin{equation} \partial_t u - \Delta u = 0, x\in W; \end{equation} \begin{equation} u(t,x) = 0, x\in \partial W, t>0; \end{equation} \begin{equation} u(0,x) = -1, x\in W. \end{equation}

It may seem silly (as this heat process seems rather classical), but I was not able to find a neat explicit reference and formula (even for dim 2). It would help to have a heat kernel expansion (of course depending on the vertex angle $\alpha$). In general, I am looking for references to heat equations with Dirichlet boundary conditions on Euclidean cones.

Many thanks in advance!

Answer 1

Lemma 1 of Brownian motion in cones by Banuelos and Smits gives the Dirichlet heat kernel for generalized cones in $\mathbb{R}^n$ in terms of Bessel functions and spherical harmonics. The paper references page 379 of Conduction of Heat in Solids by Carslaw and Jaeger as a source of a more explicit formula in $\mathbb{R}^2$.