具振动和奇异数据的拟线性Riccati型方程
Quasilinear Riccati type equations with oscillatory and singular data
摘要：
本文讨论了拟线性Riccati型方程\begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + σ\quad \text{in} ~Ω, \\ u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*}解的存在性，其中方程具有分布或测度数据$σ$ .这里，${\rm div}\,\mathcal{A}(x, \nabla u)$是一个拟线性椭圆算子，它是在$p$ -拉普拉斯算子（$p>1$）之后建模的，而$Ω$是一个边界足够平坦的有界域（在Reifenberg意义上）。对于分布式数据，我们假设$p>1$和$q>p$。对于测量数据，我们假设它们在$Ω$、$p>\frac{3n-2}{2n-1}$和$q$中完全支持，并且$q$在次线性范围$p-1<q<1$中。在这种情况下，我们还假设$\mathcal{A}$和$\partialΩ$上有更多的正则性条件。
英文摘要：
We characterize the existence of solutions to the quasilinear Riccati type equation \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + σ\quad \text{in} ~Ω, \\ u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*} with a distributional or measure datum $σ$. Here ${\rm div}\,\mathcal{A}(x, \nabla u)$ is a quasilinear elliptic operator modeled after the $p$-Laplacian ($p>1$), and $Ω$ is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that $p>1$ and $q>p$. For measure data, we assume that they are compactly supported in $Ω$, $p>\frac{3n-2}{2n-1}$, and $q$ is in the sub-linear range $p-1<q<1$. We also assume more regularity conditions on $\mathcal{A}$ and on $\partialΩ$ in this case.
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