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On a class of graphs with prescribed mean curvature

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Abstract

We study a class of quasilinear elliptic equations on the unit ball of ℝn in the divergence form ∑ n j=1 D j{G(|x|2,|Du|2)D j u} =H(|x|) and get estimates on the boundary by using a modified barrier-function technique of Bernstein. We establish a maximum principle for the gradients of solutions and get a global gradient estimate. We prove that solutions with constant boundary condition must be radial. Finally, we apply these results to graphs {(x,u(x)):x∈H n} whereu:H n→ℝ is a smooth map of then-hyperbolic spaceH n =B(0,1) with the metric\(g = \frac{{4dx^2 }}{{(1 - \left| x \right|)^2 )^2 }}\) to get the existence of graphs with radial prescribed mean curvature.

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Authors and Affiliations

  1. Mathematics Section, International Centre for Theoretical Physics, P.O. Box 586-Miramare, 34100, Trieste, Italy

    Duong Minh Duc & Isabel M. C. Salavessa

  2. Centro de Física da Matéria Condensada, Instituto Nacional de Investigação Científica, Av. Prof. Gama Pinto, 2, 1699, Lisboa Codex, Portugal

    Duong Minh Duc & Isabel M. C. Salavessa

Authors
  1. Duong Minh Duc
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  2. Isabel M. C. Salavessa
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Minh Duc, D., Salavessa, I.M.C. On a class of graphs with prescribed mean curvature. Manuscripta Math 82, 227–239 (1994). https://doi.org/10.1007/BF02567699

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