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Given a bounded domain $\Omega\subset\mathbb R^2$, and $f:\Omega\to\mathbb R$, we know that the graph of function $f$ is a surface in $\mathbb R^3$.

Suppose, we just know the Gaussian Curvature $k$ of the surface and the value/behavior of $f$ along/restriction on the boundary $\partial \Omega$.

Q: How to find the $f$?

PS: Any computational method is also welcome.

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    How is it related to covering spaces? – 2017-01-05
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    @JohnMa No idea, someone modified the tag. – 2017-01-05

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This is the problem of prescribed Gauss curvature, one of the classic examples of a PDE of Monge-Ampere type. See section "Applications" for the explicit PDE. It is a nonlinear elliptic PDE and not so easy to solve in general, though there is a lot of modern research on the subject and Monge-Ampere equations overall.