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Prescribing the Gaussian Curvature in a Subdomain of \(\mathbb {S}^2\) with Neumann Boundary Condition

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Abstract

The problem of prescribing the Gaussian curvature under a conformal change of the metric leads to the equation:

$$\begin{aligned} -\Delta u +2 = 2 K(x) e^u. \end{aligned}$$

Here we are concerned with the problem posed on a subdomain \(\Sigma \subset \mathbb {S}^2\) under Neumann boundary condition. By using min-max techniques we give a new existence result that generalizes and unifies previous work on the argument. For sign-changing \(K\), compactness of solutions is not known in full generality, and this difficulty is bypassed via an energy comparison argument.

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Acknowledgments

The authors have been supported by the Spanish Ministry of Science and Innovation under Grant MTM2011-26717 and by J. Andalucia (FQM 116).

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Authors and Affiliations

  1. Dpto. Análisis Matemático, 18071, Granada, Spain

    Rafael López-Soriano & David Ruiz

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  1. Rafael López-Soriano
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  2. David Ruiz
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Correspondence to David Ruiz.

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López-Soriano, R., Ruiz, D. Prescribing the Gaussian Curvature in a Subdomain of \(\mathbb {S}^2\) with Neumann Boundary Condition. J Geom Anal 26, 630–644 (2016). https://doi.org/10.1007/s12220-015-9566-x

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  • DOI: https://doi.org/10.1007/s12220-015-9566-x

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