Geometric estimates of~solutions of~quasilinear elliptic inequalities

A. A. Kon'kov

Izvestiya. Mathematics , Tome 84 (2020) no. 6, pp. 1056-1104

Résumé

Suppose that $p>1$ and $\alpha$ are real numbers with $p-1 \leqslant \alpha \leqslant p$. Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$, $n \geqslant 2$. We consider the inequality $$ \operatorname{div} A (x, D u)+b (x) |D u|^\alpha\geqslant 0, $$ 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 $$ C_1|\xi|^p\leqslant\xi A(x, \xi),\quad |A (x, \xi)|\leqslant C_2|\xi|^{p-1},\qquad C_1, C_2=\mathrm{const}>0, \quad p>1, $$ 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.

Détail

Keywords: non-linear operators, elliptic inequalities, boundary regularity conditions.