# A maximum principle for mean-curvature type elliptic inequalities

### James Serrin

University of Minnesota, Minneapolis, United States

## Abstract

Consider the divergence structure elliptic inequality

$div{A(x,u,Du)}+B(x,u,Du)≥0(1)$

in a bounded domain $Ω⊂R_{n}$. Here

$A(x,z,ξ):K→R_{n},B(x,z,ξ):K→R,K=Ω×R_{+}×R_{n},$

and $A$, $B$ satisfy the following conditions

$⟨A(ξ),ξ⟩≥∣ξ∣−c(x)z−a(x),∣A(x,z,ξ)∣≤Const.,B(x,z,ξ)≤b(x),$

for all $(x,z,ξ))∈K$, 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 *any solution which satisfies $u≤0$ on $∂Ω$ must be a priori bounded above in $Ω$.*

This question arises, in particular, when one is interested in the mean curvature equation

$div1+∣Du∣_{2} Du =nH(x).$

## Cite this article

James Serrin, A maximum principle for mean-curvature type elliptic inequalities. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 389–398

DOI 10.4171/JEMS/59