Weyl asymptotic law In Panoramic view in Riemannian geometry of  Berger, I met the following formula
$$\sum e^{-\lambda_i t} \sim \frac{\vert \Omega\vert }{2\pi t} -\frac{\vert \partial \Omega\vert}{\sqrt{2\pi t}} + \frac{1-r}{6}$$
where the $\lambda_i$ are the spectrum of the Laplace on a domain $\Omega$ with $r$ holes. He gives the original paper of Kac as reference, but Kac only gives a scheme of the proof for polygonal convex domain. Where can I find a full (readable) proof?
 A: I believe you can find a brief literature review and general proof in a delightful paper by McKean and Singer:

"Curvature and the Eigenvalues of the Laplacian" by H.P. McKean and I.M. Singer.
Journal of Differential Geometry 1 (1967) 43-69.

They note (as you observed) that Kac conjectured but did not prove the estimate
$$ Z(t) = \frac{\mbox{area}}{4\pi t} - \frac{\mbox{perimeter}/4}{\sqrt{4\pi t}} + \frac{1}{6}(1-h) + o(1)$$
for the heat trace $Z$ as $t\to 0$ from above.
They prove two formulas. The first is for a closed manifold. The second is for a $d$-dimensional manifold with boundary:

$$(4\pi t)^{d/t}Z = (\mbox{volume of manifold}) \pm \frac{1}{4}\sqrt{4\pi t}(\mbox{area of boundary}) + \frac{t}{3} (\mbox{integral of scalar curvature}) - \frac{t}{6}(\mbox{integral of mean curvature})$$

In case of a smooth planar domain with holes, Gauss-Bonnet applied to the double of the domain gives Kac's formula.
(The $\pm$ in the formula is $+$ for Neumann conditions and $-$ for Dirichlet conditions on the boundary.)
