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On the behavior of solutions of quasilinear elliptic inequalities near a boundary point (1904.03394v1)

Published 6 Apr 2019 in math.AP

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

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