1396 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

12

t2∫

Ω

a(x)∇v(x) ·∇v(x)dx+12

∫Ω

a(x)∇u(x) ·∇u(x)dx

+t∫

Ω

a(x)∇u(x) ·∇v(x)dx− 12

∫Ω

a(x)∇u(x) ·∇u(x)dx+ t∫

∂Ω

b(x)γv(x)dσ

(z,v)≤ 12

t∫

Ω

a(x)∇v(x) ·∇v(x)dx+∫

Ω

a(x)∇u(x) ·∇v(x)dx+∫

∂Ω

b(x)γv(x)dσ

This holds for all t > 0 and so, (z,v) ≤∫

Ωa(x)∇u(x) ·∇v(x)dx+

∫∂Ω

b(x)γv(x)dσ . Justlet t→ 0. However, this also holds for t < 0 and in this case, when we divide by t it turns theinequality around and so we find that (z,v)≥

∫Ω

a(x)∇u(x) ·∇v(x)dx+∫

∂Ωb(x)γv(x)dσ .

Now specializing to v ∈C∞c (Ω) , z =−∇(a(x)∇u(x)) because C∞

c (Ω) is dense in L2 (Ω).You could consider instead ∑i, j ai jv,iv, j where (ai j) is a positive definite matrix for examplewith a similar argument.

In the example, maybe V ={

u ∈ H1 (Ω) : γu = 0 on ΓD}

where ΓD ⊆ ∂Ω. Then ifA = ∂φ what would the evolution equation u′+Au = f ,u(0) = u0 give you? You wouldhave u = 0 on ΓD since u ∈ D(A) ⊆ V . What about on ∂Ω\ΓD ≡ ΓN? We have from theabove that for all v ∈V,∫

Ω

−∇(a(x)∇u(x))vdx =∫

Ω

a(x)∇u(x) ·∇v(x)dx+∫

∂Ω

b(x)γv(x)dσ

and so at least formally, from the divergence theorem,∫

ΓN(a(∇u ·n)+b(x))vdσ = 0

where for n the unit outer normal. Since this holds for all v it follows that aun = −bon ΓN .

42.5 Moreau’s TheoremThere is a really amazing theorem, Moreau’s theorem. It is in [24], [13] and [116]. Itinvolves approximating a convex function with one which is differentiable.

Theorem 42.5.1 Let φ be a convex lower semicontinuous proper function defined on H.Define

φ λ (x)≡miny∈H

(1

2λ|x− y|2 +φ (y)

)Then the function is well defined, convex, Frechet differentiable, and for all x ∈ H,

limλ→0

φ λ (x) = φ (x) ,

φ λ (x) increasing as λ decreases. In addition,

φ λ (x) =1

2λ|x− Jλ x|2 +φ (Jλ (x))

where Jλ x≡ (I +λ∂φ)−1 (x). The Frechet derivative at x equals Aλ x where

Aλ =1λ− 1

λ(I +λ∂φ)−1 =

1λ− 1

λJλ

Also, there is an interesting relation between the domain of φ and the domain of ∂φ

D(∂φ)⊆ D(φ)⊆ D(∂φ)