This pdf gives me the following formula for the Laplacian in polar coordinates:
$$\Delta f=D_{r_n}^2f+\frac{n-1}{r_n}D_{r_n}f+\frac{1}{r_n^2}\Lambda_nf,$$
where $n$ is the dimension of the space we are working in (i.e. $f$ has domain contained in $\mathbb{R}^n$), $\Lambda_n$ is the Laplacian on the sphere $S^{n-1}$, and the polar coordinates are $(r_n,\phi_2,\dotsc,\phi_n)$. I took a course called Higher Analysis last semester. In the course's notes, I find:
$\textbf{Proposition 1.9.24 }$ Suppose that $\Omega$ is a bounded domain, which satisfies the exterior cone condition. Then the classical Dirichlet problem is uniquely solvable in $\Omega$.
Proof. Fix a point $X\in\partial\Omega$. Since $\Omega$ satisfies the exterior cone condition, there exist constants $\alpha$ and $h$ such that the cone $\Gamma(X,\alpha,h)$ is such that $\Gamma(X,\alpha,h)\cap\overline{\Omega}=\{X\}$. We normalise our setting by assuming that $\Gamma(X,\alpha,h)\cap\left(X+{\Bbb S}^{n-1}\right)\neq\emptyset$. To construct a barrier at $X$ we argue as follows. Consider all functions $v$ of the form $$w(x)=r^\lambda\varphi_0(\omega),$$ where $x=r\omega$ (polar coordinates centred at $X$), $\lambda>0$, and $\varphi_0$ is the eigenfunction associated to the lowest eigenvalue $\mu$ of the eigenvalue problem $$\Delta_{\Bbb S^{n-1}} U=-\mu U$$ on $\Bbb S^{n-1}\setminus\Gamma(0,\alpha,h)$ with Dirichlet boundary conditions. A well known result in spectral theory, which we do not prove in these notes, asserts that $\mu>0$ and that $\varphi_0$ is a smooth function, which is strictly positive in $\Bbb S^{n-1}\setminus\Gamma(0,\alpha,h)$. By using the formula for the Laplacin $\Delta$ in polar coordinates, we find that $$\Delta w(x)=r^{\lambda-2}\varphi_0\left[\alpha^2+(n-2)\alpha-\mu\right].$$ It is straightforward to check that if $\alpha=(1/2)\left[\displaystyle\sqrt{\displaystyle (n-2)^2+4\mu}-(n-2)\right]$, then $w$ is harmonic in $\Omega$. Furthermore, $w(X)=0$, and $w>0$ in $\Omega\setminus\{X\}$. Thus, $w$ is a barrier at $X$. Hence $X$ is a regular point relative to $\Omega$.
$\quad$ Since this holds for every point in $\partial\Omega$, the classical Dirichlet problem is solvable by Lemma $\text{1.9.19}$. The uniqueness follows from the maximum principle. $\tag*{$\square$}$
Besides wondering what "well-known result in spectral theory" the teacher is referring to, I am stuck on calculating that laplacian. I mean, if I apply the pdf's formula, I seem to get, noting $r_n$ is $r$ and $\Delta_{\mathbb{S}^{n-1}}$ is $\Lambda_n$:
\begin{align*} \Delta w={}&\left(D_r^2+\frac{n-1}{r}D_r+\frac{1}{r^2}\Lambda_n\right)(r^\lambda\varphi_0)={} \\ {}={}&(D_r^2r^\lambda)\varphi_0+\left(\frac{n-1}{r}D_rr^\lambda\right)\varphi_0+\frac{1}{r^2}r^\lambda\Lambda_n\varphi_0={} \\ {}={}&\lambda(\lambda-1)r^{\lambda-2}\varphi_0+\lambda(n-1)r^{\lambda-2}\varphi_0-r^{\lambda-2}\mu\varphi_0={} \\ {}={}&r^{\lambda-2}\varphi_0(\lambda(\lambda-1)+\lambda(n-1)-\mu). \end{align*}
But that bracketed term, which is $\lambda^2+\lambda(n-2)-\mu$, is pretty different from the notes' formula. So what am I doing wrong? What am I missing?
I asked the professor, and he was pretty convinced there should be a $\lambda$ there, because of course we cannot fix $\alpha$, since $\alpha$ is given by the problem (specifically by the domain).
He also added that this Laplacian was the Laplacian on the sphere minus the cone, and that $\phi_0$ was the eigenfunction relative to the least eigenvalue of that Laplacian with Dirichlet conditions (I guess value fixed to zero) on the boundary (i.e. sphere intersected with cone).