Imagine a circular (or cylindrical, in cross-section) object being supported by an elastic string. Like this:
To actually compute the equilibrium mass-string configuration, I would have to take some values for the mass of the object and for the resistance of the string. Instead, I simply chose the position of the object: it is the unit circle with center at . It remains to find the equilibrium shape of the string. The shape is described by equation where minimizes the appropriate energy functional subject to boundary conditions and the obstacle . The functional could be the length
or its quadratization
The second one is nicer because it yields linear Euler-Lagrange equation/inequality. Indeed, the obstacle permits one-sided variations with smooth and compactly supported. The linear term of is , which after integration by parts becomes . Since the minimizer satisfies , the conclusion is whenever . Therefore, everywhere (at least in the sense of distributions), which means is a convex function. In the parts where the string is free, we can do variation of either sign and obtain ; that is, is an affine function there.
The convexity of in the part where it touches the obstacle is consistent with the shape of the obstacle: the string can assume the same shape as the obstacle.
The function can now be determined geometrically: the only way the function can come off the circle, stay convex, and meet the boundary condition is by leaving the circle along the tangents that pass through the endpoint . This is the function pictured above. Its derivative is continuous: Lipschitz continuous, to be precise.
The second derivative does not exist at the transition points. Still, the minimizer has a higher degree of regularity (Lipschitz continuous derivative) than a generic element of the function space in which minimization takes place (square-integrable derivative).
As a bonus, the minimizer of energy turns out to minimize the length as well.
All in all, this was an easy problem. Next post will be on its fourth-order version.