54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1717

Obviously you could follow identical reasoning to include many other examples ofmore complexity. What does it mean for u ∈ D(A)? It means that in a weak sense

−au+∇ · (a(x)∇u) ∈ H.

Since A is sectorial for S−a,φ for any 0 < φ < π/2, this has shown the existence of a weaksolution to the partial differential equation along with appropriate boundary conditions,

−au+∇ · (a(x)∇u) = f , u ∈V.

What are these appropriate boundary conditions? u = 0 on Γ is one. the other would bea variational boundary condition which comes from integration by parts. Letting v ∈ V,formally do the following using the divergence theorem.

( f ,v)H =∫

Ω

(−au+∇ · (a(x)∇u))vdx

=∫

Ω

−auvdx+∫

∂Ω

(a(x)∇uv) ·nds−∫

Ω

a(x)∇u(x) ·∇v(x)dx

= ( f ,v)H +∫

∂Ω\Γ(a(x)∇u) ·nvds

and so the other boundary condition is

a(x)∂u∂n

= 0 on ∂Ω\Γ.

To what extent this weak solution is really a classical solution depends on more technicalconsiderations.

54.3.3 Fractional Powers Of Sectorial OperatorsIt will always be assumed in this section that A is sectorial for the sector S−a,φ wherea > 0. To begin with, here is a useful lemma which will be used in the presentation of thesefractional powers.

Lemma 54.3.15 The following holds for α ∈ (0,1) and σ < t.∫ t

σ

(t− s)α−1 (s−σ)−α ds =π

sin(πα)

In particular, ∫ 1

0(1− s)α−1 s−α ds =

π

sin(πα).

Also for α,β > 0

Γ(α)Γ(β ) =

(∫ 1

0xα−1 (1− x)β−1 dx

)Γ(α +β ) .