A maximum principle for mean-curvature type elliptic inequalities

James Serrin

doi:10.4171/jems/59

JournalsjemsVol. 8, No. 2pp. 389–398

A maximum principle for mean-curvature type elliptic inequalities

James Serrin
University of Minnesota, Minneapolis, United States

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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 $\boldsymbol A$ , $B$ 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 $(x,z,\boldsymbol \xi)) \in K$ , 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

{\rm div}\frac{Du}{\sqrt{1+|Du|^2}} = 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