Interior curvature estimates for hypersurfaces of prescribed mean curvature
In an earlier paper [Anal. Mathém. Appl. 55-66 (1988; Zbl 0668.35028)] \textit{L. Caffarelli}, \textit{L. Nirenberg} and \textit{J. Spruck} established under some conditions a curvature bound \[ | A| v(0)\leq c(n)R^{-1}\sup_{B_ r(0)}v^ 2 \] for a graph u(x) in \({\mathbb{R}}^ n\) of prescribed mean curvature H; here \(| A|\) denotes the norm of the second fundamental form, \(v=\sqrt{1+| Du|^ 2},\quad x\in {\mathbb{R}}^{n-1}.\) The present work provides an estimate \[ | A| v(0)\leq cR^{-1}\sup_{{\mathcal H}_ R(0)}v, \] where \({\mathcal H}_ R(0)\) is a ball in \({\mathbb{R}}^ n\). In general, c depends on n, H and the first two covariant derivatives of H. However, in particular cases of interest the estimate simplifies notably. One such case is the capillary equation \(H=ku\), \(k>0\), for which the authors obtain \[ | A| v(0)\leq c(n,k)R^{-3} \] as \(R\to 0\). If \(n=2\) and \(H\equiv const\). the method yields \[ | A| v(0)\leq cR^{-1}\sup_{B_ r(0)}v \] with c independent of R, v. With regard to this latter case, the reviewer calls attention to a paper of \textit{L. Simon} [Pac. J. Math. 69, 245-268 (1976; Zbl 0354.35040)] in which a bound independent of v is given. See also \textit{F. Sauvigny} [Math. Z. (to appear)]. Explicit bounds in this case were given by \textit{Y. Shih} [Chin. J. Math. 16, 111-122 (1988; see the preceding review)] and by the reviewer [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1987, No.2, 5-14 (1987; Zbl 0645.53003)]. As pointed out in the last reference, an analogous estimate for \(| D^ 2u(0)|\) would not be possible, although such a bound does hold with c a function of \(| v(0)|\) alone and the sup v term not appearing.
scientific article; zbMATH DE number 4119110
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