Quantitative stratification of $F$-subharmonic functions

In this paper, we study the singular sets of $F$-subharmonic functions $u: B_{2}(0^{n})\rightarrow\mathbf{R}$, where $F$ is a subequation. The singular set $\mathcal{S}(u)\subset B_{2}(0^{n})$ has a stratification $\mathcal{S}^{0}(u)\subset\mathcal{S}^{1}(u)\subset\cdots\subset\mathcal{S}^{k}(u)\subset\cdots\subset\mathcal{S}(u)$, where $x\in\mathcal{S}^{k}(u)$ if no tangent function to $u$ at $x$ is $(k+1)$-homogeneous. We define the quantitative stratifications $\mathcal{S}_{\eta}^{k}(u)$ and $\mathcal{S}_{\eta,r}^{k}(u)$ satisfying $\mathcal{S}^{k}(u)=\cup_{\eta}\mathcal{S}_{\eta}^{k}(u)=\cup_{\eta}\cap_{r}\mathcal{S}_{\eta,r}^{k}(u)$.

When homogeneity of tangents holds for $F$, we prove that $dim_{H}\mathcal{S}^{k}(u)\leq k$ and $\mathcal{S}(u)=\mathcal{S}^{n-p}(u)$, where $p$ is the Riesz characteristic of $F$. And for the top quantitative stratification $\mathcal{S}_{\eta}^{n-p}(u)$, we have the Minkowski estimate $\text{Vol}(B_{r}(\mathcal{S}_{\eta}^{n-p}(u)\cap B_{1}(0^{n})))\leq C\eta^{-1}(\int_{B_{1+r}(0^{n})}\Delta u)r^{p}$.

When uniqueness of tangents holds for $F$, we show that $S_{\eta}^{k}(u)$ is $k$-rectifiable, which implies $\mathcal{S}^{k}(u)$ is $k$-rectifiable.

When strong uniqueness of tangents holds for $F$, we introduce the monotonicity condition and the notion of $F$-energy. By using refined covering argument, we obtain a definite upper bound on the number of $\{\Theta(u,x)\geq c\}$ for $c\gt 0$, where $\Theta(u,x)$ is the density of $F$-subharmonic function $u$ at $x$.

Geometrically determined subequations $F(\mathbb{G})$ are a very important type of subequation (when $p=2$, homogeneity of tangents holds for $F(\mathbb{G})$; when $p\gt 2$, uniqueness of tangents holds for $F(\mathbb{G})$). By introducing the notion of $\mathbb{G}$-energy and using quantitative differentation argument, we obtain the Minkowski estimate of quantitative stratification $\text{Vol}(B_{r}(\mathcal{S}_{\eta,r}^{k}(u))\cap B_{1}(0^{n}))\leq Cr^{n-k-\eta}$.

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Received 20 September 2017

Accepted 9 January 2019

Published 11 January 2022