Polyharmonic boundary value problems

2.4 Hilbert space theory 35
To the linear differential operator A defined by
we associate the bilinear form
Ψ(u,v) = (u,v) +∑(−1) ◦
|β|
Au := (−∆) m u +∑ D
◦
β [aβ,µ(x)D µ u] (2.28)
Ω
a β,µ(x)D µ uD β vdx for all u,v ∈ H m (Ω), (2.29)
where (., .) is defined in (2.13). Formally, Ψ is obtained by integrating by parts
Auv and by neglecting the boundary integrals. We point out that, in view of (2.27),
Ψ(u,v) is well-defined for all u,v ∈ Hm (Ω).
Let us recall that m j denotes the highest order derivatives of u appearing in B j.
With no loss of generality, we may assume that the boundary conditions (2.25) are
ordered for increasing m j’s so that
m j ≤ m j+1 for all j = 1,...,m − 1. (2.30)
Moreover, we assume that the coefficients in (2.25) satisfy
b j,α ∈ C 2m−m j (Ω) for all j = 1,...,m and |α| ≤ m j; (2.31)
by this, we mean that the functions b j,α are restrictions to the boundary ∂Ω of
functions in C 2m−m j(Ω).
We also need to define well-behaved systems of boundary operators.
Definition 2.10. Let k ∈ N + . We say that the boundary value operators {Fj(x;D)} k j=1
satisfying (2.30) form a normal system on ∂Ω if mi < m j whenever i < j and if
Fj(x;D) contains the term ∂ m j/∂ν m j with a coefficient different from 0 on ∂Ω.
Moreover, we say that {Fj(x;D)} k j=1 is a Dirichlet system if, in addition to the
above conditions, we have m j = j − 1 for j = 1,...,k; the number k is then called
the order of the Dirichlet system.
Remark 2.11. The assumption “Fj contains the term ∂ m j/∂ν m j with a coefficient
different from 0 on ∂Ω” requires some explanations since it may happen that the
term ∂ m j/∂ν m j does not appear explicitly in Fj. One should then rewrite the boundary
conditions on ∂Ω in local coordinates; the system of coordinates should contain
the n − 1 tangential directions and the normal direction ν. Then the assumption is
that in this new system of coordinates the term ∂ m j/∂ν m j indeed appears with a
coefficient different from 0. For instance, imagine that m j = 2 and that ∆u represents
the terms of order 2 in Fj; it is known that if ∂Ω and u are smooth, then
∆u = ∂ 2u ∂ν 2 + (n − 1)H ∂u
∂ν + ∆τu on ∂Ω, where H denotes the mean curvature at the
boundary and ∆τu denotes the tangential Laplacian of u. Therefore, any boundary
operator which contains ∆ as principal part satisfies this condition.
It is clear that if a normal system of boundary value operators {Fj(x;D)} k j=1 is
such that mk = k − 1, then it is a Dirichlet system.