40.1 A Fundamental Inequality
The proof of Korn’s inequality depends on a fundamental inequality involving negative
Sobolev space norms. The theorem to be proved is the following.
Theorem 40.1.1 Let f ∈ L2
Ω is a bounded Lipschitz domain. Then there
exist constants, C1 and C2 such that
0,2,Ω represents the L2 norm and
−1,2,Ω represents the norm in the
dual space of H01
, denoted by H−1
Similar conventions will apply for any domain in place of Ω. The proof of this theorem
will proceed through the use of several lemmas.
Lemma 40.1.2 Let U− denote the set,
where g : ℝn−1 → ℝ is Lipschitz and denote by U+ the set
Let f ∈ L2
and extend f to all of ℝn in the following way.
Then there is a constant, Cg, depending on g such that
Proof: Let ϕ ∈ Cc∞
Consider the first integral on the right in 40.1.1. Changing the variables, letting
yn = 2g
in the first term of the integrand and 3g
in the next, it
it follows ψ = 0 when yn = g
Now from the definition of ψ given above,
It remains to establish a similar inequality for the case where the derivatives are taken
with respect to xi for i < n. Let ϕ ∈ Cc∞
Changing the variables as before, this last integral equals
Using this in 40.1.3, the integrals in this expression equal
and so ϕ− 3ψ1 + 2ψ2 ∈ H01
It also follows from the definition of the functions, ψi
and the assumption that g
is Lipschitz, that
where Cg is a constant which depends on g. This inequality along with 40.1.2
follows from 40.1.4 and the equation,
which results in the same way as before by changing variables using the definition of f off
U−. This proves the lemma.
The next lemma is a simple application of Fourier transforms.
Lemma 40.1.3 If f ∈ L2
, then the following formula holds.
Proof: For ϕ ∈ Cc∞
is an equivalent norm to the usual Sobolev space norm for H01
and is used in the
following argument which depends on Plancherel’s theorem and the fact that