I have been trying to analyse the shapes that in hydrostatics when surface tension and gravity balance each other. The theory in the absence of gravity is well studied, which leads to constant mean curvature problem and shapes like the catenoid. I wanted to know if there is a way to solve the same equation in the presence of gravity. Specifically, if I have this equation, $$ \nabla p = \rho g $$ and $$ p = \gamma C$$ where C is the curvature of the surface, where I've been using the definition of curvature of a monge patch $z = z(x,y)$. I know how to solve the system for axisymmetric surface numerically. Is there a method or a transformation that helps to solve the system in 3d? A reference towards numerical methods of solving arbitrary curvature equations would also help. Thanks!
2026-04-02 17:01:21.1775149281
Variable mean curvature equation
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