Polyharmonic boundary value problems

2.3 Boundary conditions 31
∂Ω is needed. Let us also mention that there is a simple way to remember the embeddings
in Theorem 2.6. It is based on the so-called regularity index, see [11, Section
8.7]. In n-dimensional bounded domains Ω, the regularity index for W m,p (Ω)
is m − n/p whereas for C k,γ (Ω) it is k + γ. A Sobolev space is embedded into any
other space with a smaller regularity index. For instance, W m,p (Ω) ⊂ W µ,q (Ω)
provided m − n/p ≥ µ − n/q (and m ≥ µ). Also W m,p (Ω) ⊂ C k,γ (Ω) whenever
m − n/p ≥ k + γ and γ ∈ (0,1), which is precisely the statement in Theorem 2.6. A
similar rule is also available for trace operators, namely if m − n/p ≥ µ − (n − 1)/q
(and m > µ) then the trace operator on W m,p (Ω) is continuous into W µ,q (∂Ω).
We conclude this section with the multiplicative properties of functions in
Sobolev spaces.
Theorem 2.7. Assume that Ω ⊂ R n is a Lipschitzian domain. Let m ∈ N + and p ∈
[1,∞) be such that mp > n. Then W m,p (Ω) is a commutative Banach algebra.
Remark 2.8. Theorem 2.7 can be generalised by considering multiplications of
functions in possibly different Sobolev spaces. For instance, if m1,m2 ∈ N + and
µ = min{m1,m2,m1 + m2 − [ n 2 ] − 1}, then Hm1(Ω)H m2(Ω) ⊂ H µ (Ω).
We postpone further properties of the Hilbertian critical embedding, that is, H m ⊂
L 2n/(n−2m) with n > 2m, to Sections 7.3 and 7.8. The reasons are both that we need
further tools and that these properties have a natural application to nonexistence
results for semilinear polyharmonic equations at critical growth.
2.3 Boundary conditions
For the rest of Chapter 2, we assume the domain Ω to be bounded. Under suitable
assumptions on ∂Ω, to equation (2.2) we may associate m boundary conditions.
These conditions will be expressed by linear differential operators B j(x;D), namely
B j(x;D)u = h j for j = 1,...,m on ∂Ω, (2.14)
where the functions h j belong to suitable functional spaces. Each B j has a maximal
order of derivatives m j ∈ N and the coefficients of the derivatives are sufficiently
smooth functions on ∂Ω. The regularity assumptions on these coefficients and on
∂Ω will be made precise in each statement.
For the problems considered in this monograph, it always appears that
m j ≤ 2m − 1 for all j = 1,...,m. (2.15)
Therefore, we shall always assume that (2.15) holds, although some of our statements
remain true under less restrictive assumptions. The meaning of (2.14) will
remain unclear until the precise definition of solution to (2.2) will be given; in most
cases, they should be seen as traces, namely satisfied in a generalised sense given
by the operators (2.4).