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 $g=u^{-α}\bar g$, where $u$ is a boundary defining function, $α\in C^1(M;[0,2))$ and $\bar g$ is a $C^1$ Riemannian metric up to $M$. Since $α<2$, 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 $α$ varies along $M$. If the maximum $α\_{\mathrm{max}}$ of $α$ on $M$ is strictly larger than the critical value $α\_c=\frac{2}{n+1}$, then we prove that the points where $α$ is close to $α\_{\mathrm{max}}$ govern the leading term in the Weyl asymptotics. If $α\_{\mathrm{max}}\leqα\_c$, then the leading term is governed by the truncated volume $\vol\_g(\{\dist(\cdot,M)>λ^{-1/2}\})$. When the maximum set of $α$ 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 $α\_c$ between a boundary-dominated non-classical regime and a truncated-volume regime.