Polyharmonic boundary value problems

2.3 Boundary conditions 33
Assumptions on the boundary conditions. Assume that, according to Definition
2.9,
the linear boundary operators B j’s satisfy the complementing condition. (2.19)
We now discuss the main boundary conditions considered in this monograph.
Dirichlet boundary conditions. In this case, B j(x,D)u = B ′ j (x,D)u = ∂ j−1 u
∂ν j−1 for
j = 1,...,m so that m j = j − 1 and (2.14) become
u = h1, ..., ∂ m−1 u
∂ν m−1 = hm on ∂Ω. (2.20)
Hence, B ′ j (x;τ + tν) = t j−1 and, as mentioned in [5, p. 627], the complementing
condition is satisfied for (2.20).
Navier boundary conditions. In this case, B j(x,D)u = B ′ j (x,D)u = ∆ j−1 u for
j = 1,...,m so that m j = 2( j − 1) and (2.14) become
u = h1, ..., ∆ m−1 u = hm on ∂Ω. (2.21)
Under these conditions, if A has a suitable form then (2.2) may be written as a system
of m Poisson equations, each one of the unknown functions satisfying Dirichlet
boundary conditions. Therefore, the complementing condition follows by the theory
of elliptic systems [6].
Mixed Dirichlet-Navier boundary conditions. We make use of these conditions
in Section 5.2. They are a suitable combination of (2.20)-(2.21). For instance, if m
is odd, they read B j(x,D)u = ∂ j−1u ∂ν j−1 for j = 1,...,m − 1 and Bm(x,D)u = ∆ (m−1)/2u. Again, the complementing condition is satisfied.
Steklov boundary conditions. We consider these conditions only for the biharmonic
operator. Let a ∈ C0 (∂Ω) and to the equation ∆ 2u = f in Ω we associate the
boundary operators B1(x,D)u = u and B2(x,D)u = ∆u − a ∂u
∂ν . Then (2.14) become
u = h1 and ∆u − a ∂u
∂ν = h2 on ∂Ω. (2.22)
Since B ′ j (for j = 1,2) is the same as for (2.21), also (2.22) satisfy the complementing
condition.
More generally, Hörmander [230] characterises all the sets of boundary operators
B j which satisfy the complementing condition.
We conclude this section by giving an example of boundary conditions which do
not satisfy the complementing condition. Consider the fourth order problem