2358 CHAPTER 69. GELFAND TRIPLES

There exists p≥ 2, p < ∞ and α ∈ (0,∞) such that for all s ∈ R

sΨ(s)≥ α |s|p− c (69.1.5)

There exist c3,c4 ∈ (0,∞) such that for all s ∈ R

|Ψ(s)| ≤ c4 + c3 |s|p−1 (69.1.6)

Note that 69.1.6 implies that if v ∈ Lp (D) , Then∫D|Ψ(v)|p

′dx≤C

∫D

(1+ |v|p

′(p−1))

dx =C∫

D(1+ |v|p)dx < ∞.

Thus for v ∈ Lp (D) ,Ψ(v) is something you can do ∆ to and obtain something in V ′. Theporous medium operator A : V →V ′ is given as follows.

⟨Av,w⟩V ′,V ≡ ⟨∆Ψ(v) ,w⟩V ′,V ≡−∫

DΨ(v)wdx

What are the properties of A?

⟨A(u+λv) ,w⟩ ≡ −∫

DΨ(u+λv)wdx

and this is easily seen to be a continuous function of λ Thus A is Hemicontinuous.

⟨A(u)−A(v) ,u− v⟩ ≡ −∫

DΨ(u)(u− v)dx+

∫D

Ψ(v)(u− v)dx≤ 0

Thus −A is monotone. Also there is a coercivity estimate which is routine.

⟨A(v) ,v⟩ ≡ −∫

DΨ(v)v≤

∫D

c−α |v|p dx =C−α ||v||pV

This operator also has a boundedness estimate.

||A(v)||V ′ ≡ sup||w||V≤1

|⟨A(v) ,w⟩| ≡ sup||w||V≤1

∣∣∣∣∫DΨ(v)w

∣∣∣∣≤ sup||w||V≤1

(∫D

(c4 + c3 |v|p−1

)wdx

)

≤(∫

DC (1+ |v|p)dx

)1/p′

≤C+C(∫

D|v|p dx

)1/p′

= C+C ||v||p/p′V =C+C ||v||p−1

V .

Since Ψ is continuous, it will also follow that A is B (V ) measurable. Consider

u→ ⟨Au,w⟩ ≡ −∫

DΨ(u)wdx