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arXiv 1806.01987math.AP
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平面非齐次$\infty$ -Laplace方程的Sobolev正则性

Some sharp Sobolev regularity for inhomogeneous $\infty$-Laplace equation in plane

Herbert Koch, Yi Ru-Ya Zhang, Yuan Zhou

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摘要：

假设$Ω\Subset \mathbb R^2$和$f\in BV_{loc}(Ω)\cap C^0(Ω)$与$Ω$中的$|f|>0$。设$u\in C^0(Ω)$是非齐次$\infty$-Laplace方程$$ -Δ_{\infty} u :=-\frac12\sum_{i=1}^2(|Du|^2)_iu_i= -\sum_{i,j=1}^2u_iu_ju_{ij} =f \quad {\rm in}\ Ω. $$的粘性解.本文证明了下列结果. (i)对于$ α> 3/2$，我们有$|Du|^α\in W^{1,2}_{loc}(Ω)$，当$ α\to 3/2$时，它（渐近）尖锐。实际上，函数$w(x_1,x_2)=-x_1^ {4/3} $是$\mathbb R^2$中$-Δ_\infty w=\frac{4^3}{3^4}$的粘度解。对于任何$p> 2$，当$α\in(3/2,3-3/p)$时为$|Dw|^α\notin W^{1,p}_{loc}(\mathbb R^2)$。(ii)对于$ α\in(0, 3/2]$和$p\in[1, 3/(3-α))$，我们有$|Du|^α\in W^{1,p}_{loc}(Ω)$，当$p\to 3/(3-α)$时，它是尖锐的。没错，$ |Dw|^α\notin W^{1,3/(3-α)}_{loc}(\mathbb R^2)$。(iii)对于$ ε> 0$，我们有$|Du|^{-3+ε}\in L^1_{loc}(Ω)$，当$ε\to0$。是的，$|Dw|^{-3} \notin L^1_{loc}(\mathbb R^2)$。(iv)对于$ α> 0$，我们有$$-(|Du|^α)_iu_i= 2α|Du|^{ α-2}f \ \mbox{ almost everywhere in $Ω$}.$$，并给出了一些定量界.

英文摘要：

Suppose $Ω\Subset \mathbb R^2$ and $f\in BV_{loc}(Ω)\cap C^0(Ω)$ with $|f|>0$ in $Ω$. Let $u\in C^0(Ω)$ be a viscosity solution to the inhomogeneous $\infty$-Laplace equation $$ -Δ_{\infty} u :=-\frac12\sum_{i=1}^2(|Du|^2)_iu_i= -\sum_{i,j=1}^2u_iu_ju_{ij} =f \quad {\rm in}\ Ω. $$ The following are proved in this paper. (i) For $ α> 3/2$, we have $|Du|^α\in W^{1,2}_{loc}(Ω)$, which is (asymptotic) sharp when $ α\to 3/2$. Indeed, the function $w(x_1,x_2)=-x_1^ {4/3} $ is a viscosity solution to $-Δ_\infty w=\frac{4^3}{3^4}$ in $\mathbb R^2$. For any $p> 2$, $|Dw|^α\notin W^{1,p}_{loc}(\mathbb R^2)$ whenever $α\in(3/2,3-3/p)$. (ii) For $ α\in(0, 3/2]$ and $p\in[1, 3/(3-α))$, we have $|Du|^α\in W^{1,p}_{loc}(Ω)$, which is sharp when $p\to 3/(3-α)$. Indeed, $ |Dw|^α\notin W^{1,3/(3-α)}_{loc}(\mathbb R^2)$. (iii) For $ ε> 0$, we have $|Du|^{-3+ε}\in L^1_{loc}(Ω)$, which is sharp when $ε\to0$. Indeed, $|Dw|^{-3} \notin L^1_{loc}(\mathbb R^2)$. (iv) For $ α> 0$, we have $$-(|Du|^α)_iu_i= 2α|Du|^{ α-2}f \ \mbox{ almost everywhere in $Ω$}.$$ Some quantative bounds are also given.

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