2416 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

solution to 70.5.15 -70.5.21 along with other variational and stable boundary conditionsdepending on the choice of W and f ∈ L2(0,T ;V ′).

In order to carry out our existence and uniqueness proofs, we assume M and A satisfythe following for some δ > 0, λ ≥ 0.

⟨Bu,u⟩ ≥ δ2||u||2W −λ |u|2H , ⟨Bu,u⟩ ≥ 0 , ⟨Bu,v⟩= ⟨Bv,u⟩ , (70.5.37)

for B = M or A. This is the assumption that we use, and we note that 70.5.37 is a conse-quence of 70.5.26 -70.5.28 and Korn’s inequality [104].

70.5.2 An Approximate ProblemWe will use the Galerkin method. To do this, we will first regularize that subgradientmaterial. Let

ψε (r) =√|r|2 + ε

Then this is a convex, Lipschitz continuous function having bounded derivative which con-verges uniformly to ψ (r) = |r| on R. Also

|ψε (x)−ψε (y)| ≤ |x−y| ,∣∣ψ ′ε (t)∣∣≤ 1

And finally, ψ ′ε is Lipschitz continuous with a Lipschitz constant C/√

ε . Here ψ ′ε denotesthe gradient or Frechet derivative of the scalar valued function.

Our approximate problem for which we will apply the Galerkin method will be Pε

given by

v′+Mv+Au+Pu+ γ∗T F((un−g)+

)µ(∣∣vT − U̇T

∣∣)ψ′ε

(vT − U̇T

)= f in V ′, (70.5.38)

v(0) = v0 ∈ H, (70.5.39)

whereu(t) = u0 +

∫ t

0v(s)ds, u0 ∈V, (70.5.40)

Here the long operator on the left is defined in the following manner.⟨γ∗T F((un−g)+

)µ(∣∣vT − U̇T

∣∣)ψ′ε

(vT − U̇T

),w⟩

=∫

ΓC

F((un−g)+

)µ(∣∣vT − U̇T

∣∣)ψ′ε

(vT − U̇T

)·wT dS

Let R denote the Riesz map from V to V ′ defined by ⟨Ru,v⟩= (u,v)V . Then R−1 : H→V is a compact self adjoint operator and so there exists a complete orthonormal basis forH, {ek} ⊆V such that

Rek = λ kek

where λ k → ∞. Let Vn = span(e1, · · · ,en). Thus ∪nVn is dense in H. In addition ∪nVn isdense in V and {ek} is also orthogonal in V . To see first that {ek} is orthogonal in V,

0 = (ek,el)H =1

λ k(Rek,el)H =

1λ k⟨Rek,el⟩=

1λ k

(el ,ek)V