Second order obstacle problem

Imagine a circular (or cylindrical, in cross-section) object being supported by an elastic string. Like this:

Obstacle problem
Obstacle problem

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 {(0,0)}. It remains to find the equilibrium shape of the string. The shape is described by equation {y=u(x)} where {u} minimizes the appropriate energy functional subject to boundary conditions {u(-2)=0=u(2)} and the obstacle {u(x)\le -\sqrt{1-x^2}}. The functional could be the length

\displaystyle  L(u) = \int_{-2}^2 \sqrt{1+u'(x)^2}\,dx

or its quadratization

\displaystyle E(u) = \frac12 \int_{-2}^2 u'(x)^2 \,dx

The second one is nicer because it yields linear Euler-Lagrange equation/inequality. Indeed, the obstacle permits one-sided variations {u+\varphi} with { \varphi\le 0} smooth and compactly supported. The linear term of {E(u+\varphi)} is {\int u'\varphi'}, which after integration by parts becomes {-\int u'' \varphi}. Since the minimizer satisfies {E(u+\varphi)-E(u)\ge 0}, the conclusion is {\int u'' \varphi \le 0 } whenever {\varphi\le 0}. Therefore, {u''\ge 0} everywhere (at least in the sense of distributions), which means {u} is a convex function. In the parts where the string is free, we can do variation of either sign and obtain {u''=0}; that is, {u} is an affine function there.

The convexity of {u} 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 {u} 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 {(\pm 2,0)}. This is the function pictured above. Its derivative is continuous: Lipschitz continuous, to be precise.

First derivative is Lipschitz continuous
First derivative is Lipschitz continuous

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 {E} turns out to minimize the length {L} as well.

All in all, this was an easy problem. Next post will be on its fourth-order version.