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!



Reference for a Heat Process in a Wedge
Lemma 1 of [Brownian motion in cones](http://www.math.purdue.edu/~banuelos/Papers/cones.pdf#page=6) 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](https://books.google.com/books?id=y20sAAAAYAAJ&dq=conduction+of+heat+in+solids&hl=en&sa=X&ved=0ahUKEwiZ0I6zp9_JAhVH7YMKHWazALAQ6AEIJjAA) by Carslaw and Jaeger as a source of a more explicit formula in $\mathbb{R}^2$.


