JournalsjemsVol. 8 , No. 2DOI 10.4171/jems/59

A maximum principle for mean-curvature type elliptic inequalities

  • James Serrin

    University of Minnesota, Minneapolis, United States
A maximum principle for mean-curvature type elliptic inequalities cover

Abstract

Consider the divergence structure elliptic inequality

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

in a bounded domain Ω\RRn\Omega\subset \RR^n. Here

A(x,z,ξ):K\RRn;B(x,z,ξ):K\RR,K=Ω×\RR+×\RRn,\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 A\boldsymbol A, BB satisfy the following conditions

A(ξ),ξξc(x)za(x),A(x,z,ξ)\mboxConst.,B(x,z,ξ)b(x),\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,ξ))K(x,z,\boldsymbol \xi)) \in K, where a(x)a(x), b(x), c(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 u0u\le 0 on Ω\partial\Omega must be a priori bounded above in Ω\Omega.} This question arises, in particular, when one is interested in the mean curvature equation

divDu1+Du2=nH(x).{\rm div}\frac{Du}{\sqrt{1+|Du|^2}} = nH(x).