论文标题
$ 1 <p \ leq 2- \ frac {1} {n} $的通用潜在估计值
Universal potential estimates for $1<p\leq 2-\frac{1}{n}$
论文作者
论文摘要
我们将Kuusi-Mingione类型的所谓通用潜力估计（J.Funct。Anal。2012）扩展到单个情况$ 1 <p \ leq 2-1/n $，用于Quasilinear方程，并带有测量数据 \ begin {equation*} - \ operatorname {div}（a（x，\ nabla u））=μ \ end {equation*}在有限的$ \ mathbb {r}^n $，$ n \ geq 2 $的有限的子集$ω$中，有有限的签名度量$μ$ in $ω$。运算符$ \ operatorAtOrName {div}（a（x，x，\ nabla u））$是在$ p $ -laplacian $Δ_pu：= {\ rm div} \，（| \ nabla u |^u |^u |^{p-2} \ nabla u |假定\ mathbb {r}^n $）可以满足订单$ p $的自然增长和单调性条件，以及$ x $ - 可变的某些其他规律性条件。
We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla u))=μ \end{equation*} in a bounded open subset $Ω$ of $\mathbb{R}^n$, $n\geq 2$, with a finite signed measure $μ$ in $Ω$. The operator $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian $Δ_p u:= {\rm div}\, (|\nabla u|^{p-2}\nabla u)$, where the nonlinearity $A(x, ξ)$ ($x, ξ\in \mathbb{R}^n$) is assumed to satisfy natural growth and monotonicity conditions of order $p$, as well as certain additional regularity conditions in the $x$-variable.
