Geometric estimates of solutions of quasilinear elliptic inequalities,Izvestiya: Mathematics

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Geometric estimates of solutions of quasilinear elliptic inequalities
Izvestiya: Mathematics ( IF 0.8 ) Pub Date : 2020-12-30 , DOI: 10.1070/im8974
A. A. Kon’kov ₁

Affiliation

Suppose that $p>1$ and $\alpha$ are real numbers with $p-1 \le \alpha \le p$ . Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$ , $n \ge 2$ . We consider the inequality $\operatorname{div} A (x, D u)+b (x) |D u|^\alpha\ge 0,$ where $D=(\partial/\partial x_1, \partial/\partial x_2, \dots, \partial/\partial x_n)$ is the gradient operator, $A\colon \Omega \times \mathbb{R}^n \to \mathbb{R}^n$ and $b\colon \Omega \to [0, \infty)$ are certain functions and $C_1|\xi|^p\le\xi A(x, \xi),\quad |A (x, \xi)|\le C_2|\xi|^{p-1},\qquad C_1, C_2=\mathrm{const}>0, \quad p>1,$ for almost all $x \in \Omega$ and all $\xi \in \mathbb{R}^n$ . We obtain estimates for solutions of this inequality using the geometry of $\Omega$ . In particular, these estimates yield regularity conditions for boundary points.

中文翻译：

拟线性椭圆不等式解的几何估计

假设 $p>1$ 和 $\alpha$ 是实数，带有 $p-1 \le \alpha \le p$ 。让 $\欧米茄$ 是一个非空开子集 $\mathbb{R}^n$ ， $n \ge 2$ 。我们考虑不等式 $\operatorname{div} A (x, D u)+b (x) |D u|^\alpha\ge 0,$ 其中 $D=(\partial/\partial x_1, \partial/\partial x_2, \dots, \partial/\partial x_n)$ 是梯度算子， $A\colon \Omega \times \mathbb{R}^n \to \mathbb{R}^n$ 并且 $b\colon \Omega \to [0, \infty)$ 是某些函数和 $C_1|\xi|^p\le\xi A(x, \xi),\quad |A (x, \xi)|\le C_2|\xi|^{p-1},\qquad C_1, C_2 =\mathrm{const}>0, \quad p>1,$ 几乎 $x \in \Omega$ 所有的 $\xi \in \mathbb{R}^n$ 。我们使用的几何来获得对这个不等式的解的估计 $\欧米茄$ 。特别是，这些估计产生边界点的规律性条件。

更新日期：2020-12-30

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