Geometric estimates of solutions of quasilinear elliptic inequalities

Suppose that $p>1$ and $\alpha$ are real numbers with $p-1 \le \alpha \le p$. Let $\Omega$ be a non-emptyopen 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 $D=(\partial/\partial x_1, \partial/\partial x_2, …, \partial/\partial x_n)$ 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 usingthe geometry of $\Omega$. In particular, these estimates yield regularity conditions for boundary points.

non-linear operators, elliptic inequalities, boundary regularity conditions