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The inclination of an H-graph. (English) Zbl 0659.53006

Calculus of variations and partial differential equations, Proc. Conf., Trento/Italy 1986, Lect. Notes Math. 1340, 40-60 (1988).
[For the entire collection see Zbl 0641.00013.]
In 1909 Bernstein proved that there is no surface \(z=u(x,y)\) of mean curvature \(H\geq H_ 0>0\) defined over a region that strictly includes a disk of radius \(1/H_ 0\). In 1965 the author proved that a stronger result is true: if \(z=u(x,y)\) has mean curvature \(H\geq H_ 0>0\) in an open disk of radius \(1/H_ 0\) then \(H=H_ 0\) and such surface is necessarily a lower hemisphere. Assuming \(H_ 0=1\), the function satisfies the equation \(div Tu=2H\leq 2\quad where\quad Tu=Du/(1+| Du|^ 2)^{1/2}\) in the disk \(x^ 2+y^ 2=r^ 2<1\). This result is a formal analogue for surfaces whose mean curvature is \(H\geq H_ 0>0\) of the Bernstein theorem for minimal surfaces. Moreover, for a surface of prescribed \(H\neq 0\) defined over a disk it is possible to have further informations with no analogue for minimal surfaces.
In particular it is possible to prove [the author and E. Giusti, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 13-31 (1977; Zbl 0343.53004)] that there exists \(R_ 0=0,5654062332...\) and a nonincreasing function A(R) in \(R_ 0<R\leq 1\) with \(A(1)=0\) such that if u(x,y) is a solution of div Tu\(=2\) in the disk \(r<R\) then \(| Du(0)| <A(R)\) is true without any bound imposed on \(| u|\). Moreover the value \(R_ 0\) cannot be improved. To prove the result the authors use as a comparison function a “capillary surface” under zero gravity conditions in a cylinder with a particular “moon” section, obtaining in view of the particular geometry a condition not only necessary but also sufficient.
The author recalls a more precise result obtained by Fei-Tsen Liang using properties of generalized solutions (in the sense of perimeter theory). Liang also proved an estimate up to the boundary of \(B_ R\), \(R>1/2\) (with other assumptions) using a new class of “moon” surfaces. Finally the author observes that the methods used give as a special case a simpler proof of an Harnack inequality due to J. Serrin [Proc. Lond. Math. Soc., III. Ser. 21, 361-384 (1970; Zbl 0199.166)]; using these new methods the effect of changes in R are taken into account.
Reviewer: M.Emmer

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization