Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that accumulates at $+\infty$. Let $N(\lambda)=\#\{k\mid \lambda_k\le \lambda\}$ be the counting function. It is well known that it holds $$ {N(\lambda)} = \lambda^{n/2}|\Omega|\omega_n + O(\lambda^{\frac{n-1}2}), $$ where $|\Omega|$ is the Lebesgue measure of $\Omega$ and $\omega_n$ is the measure of the n-dimensional unit ball.
I am looking for references on the dependency of the big-O in the above formula w.r.t. $\Omega$. The best that I could find is in [1] (see also Theorem 16.1 in [2]), from which one obtains: $$ {N(\lambda)} = \lambda^{n/2}|\Omega|\omega_n + |\Omega|O(\lambda^{\frac{n-1}2}), $$ where, for any compact $K$, the big-O is uniform on $\Omega\subset K$. However, my intuition (fuelled by the Weyl conjecture) would ask for something like $$ {N(\lambda)} = \lambda^{n/2}|\Omega|\omega_n + |\Omega|^{\frac{\textbf{n-1}}{\textbf{n}}}O(\lambda^{\frac{n-1}2}), $$ with the same kind of uniformity on the big-O.
Does anyone know of results of this type?
[1]: L. Hörmander, The spectral function of an elliptic operator, 1954.
[2]: M.A. Shubin, S.I. Andersson, _ Pseudodifferential operators and spectral theory_, 1987.
