Abstract: A more comprehensive discussion on the bifurcation problems for the shape of liquid surfaces is made in this paper. The necessary conditions for bifurcation are given, and the bifurcating solutions near bifurcation points can be obtained by perturbation technique. Finally the stability of the bifurcating states is analyzed by means of the principle of minimum potential energy.

Key words: global optimization, deviation integral, variable measure, R-convergence, finite-dimensional approximation