Volume Estimates for Singular sets and Critical Sets of Elliptic Equations with H\"older Coefficients

Consider the solutions $u$ to the elliptic equation $\mathcal{L}(u) = \partial_i(a^{ij}(x) \partial_j u) + b^i(x) \partial_i u + c(x) u= 0$ with $a^{ij}$ assumed only to be H\"older continuous. In this paper we prove an explicit bound for $(n-2)$-dimensional Minkowski estimates of singular set $\mathcal{S}(u) = \{ x \in B_1 : u(x) = |\nabla u(x)| = 0\}$ and critical set $\mathcal{C}(u) \equiv \{ x\in B_{1} : |\nabla u(x)| = 0 \}$ in terms of the bound on doubling index, depending on $c \equiv 0$ or not. Here the H\"older assumption is sharp as it is the weakest condition in order to define the critical set of $u$ according to elliptic estimates. We can also obtain an optimal improvement on Cheeger-Naber-Valtorta's volume estimates on each quantitative stratum $\mathcal{S}^k_{\eta, r}$. The main difficulty in this situation is the lack of monotonicity formula which is essential to the quantitative stratification. In our proof, one key ingredient is a new almost monotonicity formula for doubling index under the H\"older assumption. Another key ingredient is the quantitative uniqueness of tangent maps. It deserves to note that our almost monotonicity is sufficient to address all the difficulties arising from the absence of monotonicity in the analysis of differential equations. We believe the idea could be applied to other relevant study.