Some Remarkable Properties of H-graphs

It is proved that if H (u) is non-decreasing and if H (?1) 6 = H (+1), then if u(x) describes a graph over a disk B R (0), with (upward oriented) mean curvature H (u), there is a bound on the gradient jD u(0)j that depends only on R, on u(0), and on the particular function H (u). As a consequence a form of Harnack's inequality is obtained, in which no positivity hypothesis appears. The results are qualitatively best possible, in the senses that a) they are false if H is constant, and b) the dependences indicated are essential (If H (?1) = ?1; H (+1) = 1, then the dependence on u(0) can be deleted). The demonstrations are based on an existence theorem for a nonlinear boundary problem with singular data, which is of independent interest. reziume. vTqvaT, H (u) arakl ebad ia da H (?1) 6 = H (+1). damt-kicebu l ia, rom Tu u(x) aGCers B R (0) Creze gansazGvr u l i Punqciis graP iks (zeviT mimar Tul i) H (u) saSu al o simru d iT, maSin arsebobs jDu(0)j grad ient is SePaseba, romel ic damokidebu l ia mxol od R-ze, u(0)-ze da H (u) Punqciaze. Sedegis saxiT miGebu l ia Harnakis utol obis erTi nairsaxeoba, romelSic ar Pigu rir ebs dadebiTobis piroba. Sedegebi Tvisebrivad gauumJ obesebad ia im azriT, rom a) isini araa samar Tl iani, Tu H mud mivia, da b) zemoT miT iTebul i damokide-bul ebebi arsebiTia In this note, we summarize and improve in some detail the material of 1]. We consider the equation div T u = 2H(u); T u = 1 q 1 + u 2 x + u 2 y hu x ; u y i (1) whose solutions are graphs u(x; y) of mean curvature H(u). Since (1) is of elliptic type, the qualitative behavior of its solutions may be expected to emulate what happens for the Laplace equation u = 0, which is usually taken