Polyharmonic boundary value problems

1.1 Classical problems from elasticity 7
Notice that σ does not play any role for clamped boundary conditions. In this
case, after an integration by parts like in (1.7), the elastic energy (1.5) becomes
12 J(u) = (∆u)
Ω
2
− f u dx
and this functional has to be minimised over the space H2 0 (Ω).
• The physically relevant boundary value problem for the pure hinged case when
∂Ω = Γ1 reads as
∆ 2u = f in Ω,
u = ∆u − (1 − σ)κ ∂u
(1.10)
∂ν = 0 on ∂Ω.
See [141, II.18 on p. 42]. These boundary conditions are named after Steklov due
the first appearance in [379]. In this case, with an integration by parts like in (1.7)
and arguing as in (1.8), the elastic energy (1.5) becomes
12 J(u) = (∆u)
Ω
2
1 − σ
− f u dx − κ u
2 ∂Ω
2 ν dω; (1.11)
for details see the proof of Corollary 5.23. This functional has to be minimised
over the space H2 ∩ H1 0 (Ω).
• On straight boundary parts κ = 0 holds and the second boundary condition in
(1.10) simplifies to ∆u = 0 on ∂Ω. The corresponding boundary value problem
∆ 2u = f in Ω,
(1.12)
u = ∆u = 0 on ∂Ω,
is in general referred to as the one with homogeneous Navier boundary conditions,
see [141, II.15 on p. 41]. On polygonal domains one might naively expect
that (1.10) simplifies to (1.12). Unless σ = 1 this is an erroneous conclusion and
instead of κ ∂u
∂ν one should introduce a Dirac-δ-type contribution at the corners.
See Section 2.7 and [293].
1.1.3 Decomposition into second order systems
Note that the combination of the boundary conditions in (1.12) or (1.10) allows for
rewriting these fourth order problems as a second order system
−∆u = w and −∆w = f in Ω,
(1.13)
u = 0 and w = 0 on ∂Ω,
respectively
−∆u = w and −∆w = f in Ω,
u = 0 and w = −(1 − σ)κ ∂u
∂ν on ∂Ω.
(1.14)