A maximum principle for mean-curvature type elliptic inequalities

Abstract

Consider the divergence structure elliptic inequality

\[ {\rm div}\{\boldsymbol A(x,u,Du)\} + B(x,u,Du) \ge 0 \leqno (1) \]

in a bounded domain \( \Omega\subset \RR^n \). Here

\[ \boldsymbol A(x,z,\boldsymbol \xi): \,K \to \RR^n; \qquad B(x,z,\boldsymbol \xi): \,K \to \RR, \qquad K = \Omega \times \RR^+ \times \RR^n, \]

and $\mathbf{A}$, $B$ satisfy the following conditions

\[ \begin{aligned} \langle\boldsymbol A(\boldsymbol \xi) , \boldsymbol \xi \rangle \ge |\xi| - & c(x)z - a(x), \qquad |\boldsymbol A(x,z,\boldsymbol \xi)| \le \mbox{Const.},\\ & B(x,z,\boldsymbol \xi) \le b(x),\end{aligned} \]

for all $(x,z,\mathbf{\xi }))\in K$, where $a(x)$, $b(x),c(x)$ are given non-negative functions. Our interest is in the validity of the maximum principle for solutions of (1), that is, the statement that {\it any solution which satisfies $u\le 0$ on $\partial \Omega$ must be a priori bounded above in $\Omega$.} This question arises, in particular, when one is interested in the mean curvature equation