78.2. MEASURABLE SOLUTIONS TO EVOLUTION INCLUSIONS 2665

Note that from Proposition 78.1.8, if v ∈ X and Bv(0) = 0, then

⟨Ku,v⟩=∫ T

0⟨Lu,v⟩ds (78.2.13)

This is because, from the Cauchy Schwarz inequality and continuity of ⟨Bu,u⟩(·) ,

⟨Bu,v⟩(0)≤ ⟨Bu,u⟩1/2 (0)⟨Bv,v⟩1/2 (0)

and if Bv(0) = 0, then from Corollary 78.1.9, ⟨Bv,v⟩1/2 (0) = 0. From the above Proposi-tion 78.1.8, this operator K is hemicontinuous and bounded and monotone as a map fromX to X ′. Thus K +A(·,ω) is a set valued pseudomonotone map for which we can applyTheorem 78.1.12 and obtain existence theorems for measurable solutions to variational in-equalities right away, but we want to obtain solutions to an evolution equation in whichK (ω) = V and the above theorem does not apply because the sum of these two opera-tors is not coercive. Therefore, we consider another operator which, when added, willresult in coercivity. Let J : V → V ′ be the duality map for 2. Thus ||Ju||V ′ = ||u||V and⟨Ju,u⟩= ||u||2V . Then J−1 : V ′→ V also satisfies

⟨f ,J−1 f

⟩= || f ||2V ′ .

The main result in this section is based on methods due to Brezis [22] and Lions [91]adapted to the case considered here where the operator is set valued, and we considermeasurability. We define the operator M : X → X ′ by

⟨Mu,v⟩ ≡⟨Lv,J−1Lu

⟩V ′,V where as above, Lu = (Bu)′ .

Then let f be measurable into V ′. Thus, in particular, f (ω) ∈ V ′ for each ω . Consider theapproximate problem and a solution uε to

εMuε (ω)+Kuε (ω)+w∗ε (ω) = f (ω)+g(ω) , w∗ε (ω) ∈ A(uε (ω) ,ω) . (78.2.14)

Where g(ω) ∈ X ′ is given by

⟨g(ω) ,v⟩ ≡ ⟨Bv(0) ,u0 (ω)⟩

where u0 (ω) is a given function measurable into W . Now for u ∈ X , we let

A (u,ω) = εMu+Ku+A(u,ω)

Then by the assumptions on A(·,ω) , there is u∗ (ω) for which ω → u∗ (ω) is measurableinto V ′, hence measurable into X ′. Therefore, ω → A (u,ω) has a measurable selection,namely εMu+Ku+u∗ (ω) and so condition 78.1.2 is verified.

By Theorem 78.1.12, a solution to 78.2.14 will exist with both uε and w∗ε measurable ifwe can argue that the sum of the operators εM +K +A(·,ω) is coercive, since this is thesum of pseudomonotone operators. From 78.1.1

inf(∫ T

0⟨u∗,u⟩ds : u∗ ∈ A(u,ω)

)+

12

∫ T

0

⟨B′u,u

⟩≥ δ (ω)

∫ T

0∥u∥p

V ds−m(ω)

and so routine considerations show that εM +K +A(·,ω) does indeed satisfy a suitablecoercivity estimate for each positive ε . Thus we have the following existence theorem forapproximate solutions.

78.2. MEASURABLE SOLUTIONS TO EVOLUTION INCLUSIONS 2665Note that from Proposition 78.1.8, if v € X and By (0) = 0, then(Ku,v) = [ (Lu, v) ds (78.2.13)This is because, from the Cauchy Schwarz inequality and continuity of (Bu, u) (-),(Bu,v) (0) < (Bu,u)'/? (0) (By, v)'/? (0)and if Bv(0) = 0, then from Corollary 78.1.9, (By,v)'/? (0) = 0. From the above Proposi-tion 78.1.8, this operator K is hemicontinuous and bounded and monotone as a map fromX to X’. Thus K + A(-,@) is a set valued pseudomonotone map for which we can applyTheorem 78.1.12 and obtain existence theorems for measurable solutions to variational in-equalities right away, but we want to obtain solutions to an evolution equation in whichK(q@) = V and the above theorem does not apply because the sum of these two opera-tors is not coercive. Therefore, we consider another operator which, when added, willresult in coercivity. Let J: V + VY’ be the duality map for 2. Thus ||Ju||y, = ||u||, and(Ju, u) = |\u||j. Then J“! : V’ + ¥ also satisfies (f.J-'f) = lvaleaeThe main result in this section is based on methods due to Brezis [22] and Lions [91]adapted to the case considered here where the operator is set valued, and we considermeasurability. We define the operator M : X — X' by(Mu,v) = (Ly, JLB) yy where as above, Lu = (Bu)’ .Then let f be measurable into ¥’. Thus, in particular, f (@) € ¥' for each w. Consider theapproximate problem and a solution ug toeMug (@) + Kug (@) + we (@) = f(@)+8(@), we(@) CA(ue(@),@). — (78.2.14)Where g(@) € X’ is given by(g (@) ,v) = (Bv (0) uo (@))where uo (@) is a given function measurable into W. Now for u € X, we let& (u,@) = eMu+ Ku+A(u,@)Then by the assumptions on A (-, @), there is u* (@) for which @ — u* (@) is measurableinto VY’, hence measurable into X’. Therefore, @ > & (u,@) has a measurable selection,namely eMu + Ku-+u* (@) and so condition 78.1.2 is verified.By Theorem 78.1.12, a solution to 78.2.14 will exist with both ug and w; measurable ifwe can argue that the sum of the operators eM + K + A(-,q@) is coercive, since this is thesum of pseudomonotone operators. From 78.1.1int( [we ,u)ds:u "€A(uo)) + s[ "U,U) ) > 6(@ o | lull ds —m(@)and so routine considerations show that eM + K +A(-,@) does indeed satisfy a suitablecoercivity estimate for each positive €. Thus we have the following existence theorem forapproximate solutions.