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Global continuity estimates for two dimensional graphs of prescribed Gauss curvature

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Abstract.

We prove the global Hölder continuity of convex solutions u ∈ C3(Ω) of the equation of prescribed positive Gauss curvature in a bounded convex domain with ∂Ω ∈ C1,β for some β ∈ (0,1]. We also obtain better regularity for the trace of u on ∂Ω. In the special case β=1 we show that and u|∂Ω ∈ C0,2/3(∂Ω). We also investigate the global continuity of solutions in C1 domains and construct an example showing that global continuity need not hold in general convex domains.

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Correspondence to John Urbas.

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Supported by an Australian Research Council Senior Fellowship.

Mathematics Subject Classification (2000): Primary 35J60; Secondary 53A05, 53C42

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Urbas, J. Global continuity estimates for two dimensional graphs of prescribed Gauss curvature. manuscripta math. 115, 179–193 (2004). https://doi.org/10.1007/s00229-004-0487-4

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  • DOI: https://doi.org/10.1007/s00229-004-0487-4

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