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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.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
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