Polyharmonic boundary value problems

1.1 Classical problems from elasticity 3
Formula (1.1) for Jsb highlights the curvature and the arclength. A two-dimensional
analogue of this functional is the Willmore functional, which is discussed below in
Section 1.8. Note that the functional Jsb does not include a term that corresponds to
an increase in the length of the beam which would occur if the ends are fixed and
the beam would bend. That is, the function in H2 ∩ H1 0 (a,b) minimising Jsb(u) −
b
a pu dx should be an approximation for the so-called supported beam which is
free to move in horizontal directions at its endpoints.
For small deformations of a beam an approximation that takes care of stretching,
bending and a force density would be
J(u) =
b
a
12 u ′′ (x) 2 + c 2 u′ (x) 2 − p(x)u(x) dx,
where c > 0 represents the initial tension of the beam which is also fixed horizontally
at the endpoints.
The linear Euler-Lagrange equation that arises from this situation contains both
second and fourth order terms:
u ′′′′ − cu ′′ = p. (1.2)
If one lets the beam move freely at the boundary points (and in the case of zero initial
tension), one arrives at the simplest fourth order equation u ′′′′ = p. This differential
equation may be complemented with several boundary conditions.
Fig. 1.1 The depicted boundary condition for the left endpoints of these four beams is “clamped”.
The boundary conditions for the right endpoints are respectively “hinged” and “simply supported”
on the left; on the right one finds “free” and one that allows vertical displacement but fixes the
derivative by a sliding mechanism.
The mathematical formulation that corresponds to the boundary conditions in
Figure 1.1 are as follows:
• Clamped: u(a) = 0 = u ′ (a), also known as homogeneous Dirichlet boundary conditions.
• Hinged: u(b) = 0 = u ′′ (b), also known as homogeneous Navier boundary conditions.
This is not a real hinged situation since the vertical position is fixed but the
beam is allowed to slide in the hinge itself.
• Simply supported: max(u(b),0)u ′′′ (b) = 0 = u ′′ (b). In applications, when the
force is directed downwards, this boundary condition simplifies to the hinged
one u(b) = 0 = u ′′ (b). However, when upward forces are present it might happen
that u(b) > 0 and then the natural boundary condition u ′′′ (b) = 0 appears.