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Some remarkable properties of $$H$$-graphs. (English) Zbl 0898.53007
Let $$H=H(u): \mathbb{R}\to \mathbb{R}$$ be a given function with $$H'(u)\geq 0$$ and $$H(-\infty)\neq H(\infty)$$. The authors consider solutions $$u=u(x,y)$$: $$B_R(0) \to\mathbb{R}$$ of the differential equation $\text{div} \bigl[(u_x,u_y)/ \sqrt{1+u^2_x+u^2_y} \bigr]= 2H(u) \tag{*}$ and derive an estimate of the gradient of $$u$$ at the origin. Notably, the bound on the norm of the gradient at the origin depends only on the radius $$R>0$$ of the disk $$B_R(0)$$ and on $$u(0)$$. (*) means that the surface given by the graph of $$u(x,y)$$ has the upward oriented mean curvature $$H(u)$$.
The proofs are based on a comparison of $$u$$ with a certain solution of (*) in a so-called moon domain. As a consequence of this estimate, a form of Harnack’s inequality is obtained in which no positivity hypothesis appears.

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35J60 Nonlinear elliptic equations 35B45 A priori estimates in context of PDEs
##### Keywords:
$$H$$-graph; gradient estimate; Harnack inequality
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