We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds $$M\times _\varrho \mathbb {R}$$ M × ϱ R . In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to $$-\infty $$ - ∞ provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast.

中文翻译：

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翘曲产品中的渐近狄利克雷问题

我们研究了扭曲积流形 $$M\times _\varrho \mathbb {R}$$ M × ϱ R 中具有规定平均曲率 H 的 Killing 图的渐近狄利克雷问题。在论文的第一部分，我们证明了在适当的假设 H 和测地球上的 Killing 圆柱的平均曲率的情况下，在测地球上具有指定边界的 Killing 图的存在性。在这个过程中，我们获得了一个统一的内部梯度估计，改进了 Dajczer 和 de Lira 之前的结果。在第二部分中，我们解决了一大类流形中的渐近狄利克雷问题，这些流形的截面曲率允许达到 0 或 $$-\infty $$ - ∞ 条件是 H 满足关于 M 的截面曲率的某些界限和 Killing 向量场的范数。