When reading the Wikipedia page on surface tension, I came across this statement:
As a result of surface area minimization, a surface will assume the smoothest shape it can (mathematical proof that "smooth" shapes minimize surface area relies on use of the Euler–Lagrange equation). Since any curvature in the surface shape results in greater area, a higher energy will also result.
I am confused by the sentence inside parentheses. I thought that solutions to the Euler-Lagrange equation are required to be smooth; it has to be separately proved that the minimal solution is smooth (how?), before applying the Euler-Lagrange equation.