The two term Weyl's conjecture states that $$N(\lambda)\sim\frac{\operatorname{area}(\Omega)}{4\pi}\lambda-\frac{\operatorname{perimeter}(\partial\Omega)}{4\pi}\sqrt\lambda$$ where $\Omega$ is a bounded domain in $\mathbb{R}^2$, and $N(\lambda)$ is the number of eigenvalues (with multiplicity) $\leq\lambda$ in the Dirichlet eigenvalue problem for the Laplacian.
I know we can get the leading term for rectangles using the explicitly known eigenvalues and Gauss circle problem and then, for Jordan-measurable regions using min-max theorems. How would I go about finding the second term, in Euclidean domains?
I think this has been proved for certain manifolds by Victor Ivrii. I am looking for some simpler proofs in $\mathbb{R}^2$.
I know how to do it for a square: we can write, $$A_2(\lambda)=4N(\lambda)-2A_1(\lambda)-1,$$ where $A_k(\lambda)=$ number of lattice points inside the $k-1$-sphere of radius $\sqrt{\lambda}$. The result follows, since $A_1(\lambda)=2\lambda^{1/2}+O(1)$ and $A_2(\lambda)=\pi\lambda+O(\lambda^{1/2}).$
A similar argument works for cubes in $\mathbb{R}^3$. Which regions can I generalize this to further?
I found this paper that discusses this for regions in which separation of variable is applicable and similar papers for circle, elliptic and polygonal domains. But they are all in Russian. Are there any references for this in English?
This is a sequel to this question.
