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arXiv 1904.03394math.AP
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拟线性椭圆型不等式解在边界点附近的性态

On the behavior of solutions of quasilinear elliptic inequalities near a boundary point

A. A. Kon'kov

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摘要：

假设$p > 1$和$p - 1 \le α\le p$是实数，并且$Ω$是${\mathbb R}^n$的非空开子集，$n \ge 2$。本文考虑不等式$$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge 0, $$，其中$D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$是梯度算子，$A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$和$b : Ω\to [0, \infty)$是对几乎所有$x \in Ω$和所有$ξ\in {\mathbb R}^n$都有$$ C_1 |ξ|^p \le ξ A (x, ξ), \quad |A (x, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 = const > 0, $$的函数.对于这个不等式的解决方案，我们得到的估计依赖于$Ω$的几何形状。特别地，这些估计暗示了边界点的正则性条件。

英文摘要：

Assume that $p > 1$ and $p - 1 \le α\le p$ are real numbers and $Ω$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge 0, $$ where $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ is the gradient operator and $A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$ and $b : Ω\to [0, \infty)$ are some functions with $$ C_1 |ξ|^p \le ξ A (x, ξ), \quad |A (x, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 = const > 0, $$ for almost all $x \in Ω$ and for all $ξ\in {\mathbb R}^n$. For solutions of this inequality we obtain estimates depending on the geometry of $Ω$. In particular, these estimates imply regularity conditions of a boundary point.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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