Weyl asymptotics for singular metrics with a variable boundary degeneracy exponent

We consider a compact smooth manifold $X$ of dimension $n+1$ with boundary $M=\partial X$. In a collar neighborhood of $M$, we assume that the metric has the form $g=u^{-\alpha}\bar g$, where $u$ is a boundary defining function, $\alpha\in C^1(M;[0,2))$ and $\bar g$ is a $C^1$ Riemannian metric up to $M$. Since $\alpha<2$, the boundary lies at finite $g$-distance and $(X,g)$ is a singular metric space. We study the Weyl asymptotics of the Friedrichs Laplacian $\triangle\_g$ when the degeneracy exponent $\alpha$ varies along $M$. If the maximum $\alpha\_{\mathrm{max}}$ of $\alpha$ on $M$ is strictly larger than the critical value $\alpha\_c=\frac{2}{n+1}$, then we prove that the points where $\alpha$ is close to $\alpha\_{\mathrm{max}}$ govern the leading term in the Weyl asymptotics. If $\alpha\_{\mathrm{max}}\leq\alpha\_c$, then the leading term is governed by the truncated volume $\vol\_g(\{\dist(\cdot,M)>\lambda^{-1/2}\})$. When the maximum set of $\alpha$ is Morse-Bott, we compute the associated constants and the logarithmic corrections. To the best of our knowledge, this is the first Weyl law in this setting with a boundary-dependent degeneracy exponent. The results highlight a sharp transition at $\alpha\_c$ between a boundary-dominated non-classical regime and a truncated-volume regime.