Topics In Analysis

23.8 Perturbation Theorems

In this section gives surjectivity of the sum of a pseudomonotone set valued map with a linear maximal monotone map and also with another maximal monotone operator added in. It generalizes the surjectivity results given earlier because one could have 0 for the maximal monotone linear operator. The theorems developed here lead to nice results on evolution equations because the linear maximal monotone operator can be something like a time derivative and X can be some sort of an L^p space for functions having values in a suitable Banach space. This is presented later in the material on Bochner integrals.

The notation

_{V ^′,V} will mean z^∗ in this section. We will not worry about the order either. Thus

〈u,z∗〉 ≡ z∗(u) ≡ 〈z∗,u〉

This is just convenient in writing things down. Also, it is assumed that all Banach spaces are real to simplify the presentation. It is also usually assumed that the Banach spaces are reflexive. Thus we can regard

′ (V × V′)= V ′ × V

and

≡ + . It is known [?] that for a reflexive Banach space, there is always an equivalent strictly convex norm. It is therefore, assumed that the norm for the reflexive Banach space is strictly convex.

There is an interesting description of L^∗ in terms of L which will be quite useful.

Proof: Let

∈G. This means that

|〈Ly,x〉| ≤ C ∥y∥ for all y ∈ D (L )

and

= for all y ∈ D. Let ∈G. Then τ = . Then

〈(x,L∗x) ,(− Ly,y)〉 = 〈x,− Ly〉+ 〈L∗x,y〉 = − 〈x,Ly〉+ 〈x,L∗y〉 = 0

Thus G

⊆^⊥. Next suppose ∈^⊥. This means that if ∈G, then

〈(x,y∗),(− Lu,u)〉 ≡ 〈x,− Lu〉 +〈y∗,u〉 = 0

and so for all u ∈ D

,

〈y∗,u〉 = 〈x,Lu〉

and so x ∈ D

. Hence for all u ∈ D,

∗ ∗ 〈y ,u〉 = 〈x,Lu 〉 = 〈L x,u〉

Then, since D

is dense, it follows that y^∗ = L^∗x and so ∈G. Thus these are the same. ■

Theorem 23.5.4 is a very nice surjectivity result for set valued pseudomonotone operators. We recall what it said here. Recall the meaning of coercive.

{ } lim inf 〈z∗,v〉: z∗ ∈ Tv = ∞ ∥v∥→∞ ||v||

In this section, we use the convenient notation

_{V ^′,V} ≡ z^∗.

Recall the definition of pseudomonotone.

Also recall the definition of modified bounded pseudomonotone. It is just the above except that the limit condition is replaced with the following condition: If u_i → u weakly in X and

∗ lim siu→p∞ 〈ui,ui − u〉 ≤ 0, (23.8.90)

(23.8.90)

then there exists a subsequence, still denoted as

such that for each v ∈ X, there exists u^∗ ∈ Au such that

∗ ∗ lim iin→f∞ 〈u i,ui − v〉 ≥ 〈u (v),u− v〉. (23.8.91)

(23.8.91)

Also recall that this more general limit condition along with the assumption 1 and the assumption that A is bounded is sufficient to obtain condition 2. This was Lemma 23.4.9 proved earlier and stated here for convenience.

Proof: Is A bounded? Of course, because the norm of X is stronger than the norm on V . Is Au convex and closed? This also follows because X ⊆ V . It is clear that Au is convex. If

⊆ Au and z_n → z in X^′, then does it follow that z ∈ Au? Since A is bounded, there is a further subsequence which converges weakly to w in V ^′. However, Au is convex and closed so it is weakly closed. Hence w ∈ Au and also w = z. It only remains to verify the pseudomonotone limit condition. Suppose then that u_n → u weakly in X and for z_n^∗∈ Au_n,

lim sup 〈z∗n,un − u〉 ≤ 0 n→∞

Then it follows that Lu_n → Lu weakly in V ^′ and u_n → u weakly in V so u ∈ X. Hence the assumption that A is L pseudomonotone implies that for every v ∈ V, and for every v ∈ X, there exists z^∗

∈ Au ⊆ V ^′⊆ X^′ such that

∗ ∗ lim nin→f∞ 〈zn,un − v〉 ≥ 〈z (v),u− v〉

The last claim goes the same way. You just have to take a subsequence. ■

Then we have the following major surjectivity result. In this theorem, we will assume for simplicity that all spaces are real spaces. Versions of this appear to be due to Brezis [?] and Lions [?]. Of course the theorem holds for complex spaces as well. You just need to use Re

instead of .

Proof: Let F be the duality map for p = 2. Consider the Banach space X given by

X = D (L), ∥u∥X ≡ ∥u ∥V + ∥Lu ∥V ′

This is isometric with the graph of L with the graph norm and so X is reflexive. Now define a set valued map G_ε on X as follows. z^∗∈ G_ε

means there exists w^∗∈ Au such that.

〈z∗,v〉 ′ = ε〈Lv,F −1(Lu)〉 ′ + 〈Lu, v〉 ′ + 〈w∗,v〉 ′ X,X V ,V V ,V V ,V

It follows from Lemma 23.8.7 that G_ε is the sum of a set valued L modified bounded pseudomonotone operator with an operator which is demicontinuous, bounded, and monotone, hence pseudomonotone. Thus by Lemma 23.5.2 it is L modified bounded pseudomonotone. Is it coercive?

{ 〈 〉 } 〈z∗,u〉 +ε Lu,F−1 (Lu )V ′,V + 〈Lu,u〉V′,V ∗ ∥ul∥im→ ∞inf ----------------||u||-----------------: z ∈ Au = ∞? X X

It equals

{ ∗ 〈 − 1 − 1 〉 } lim inf 〈z,u〉+-ε--FF---(Lu),F---(Lu-)V-′,V +-〈Lu,u〉V′,V-: z∗ ∈ Au ∥u∥X→ ∞ ||u ||X

and this is

because L is monotone. Now let M be an arbitrary positive number. By assumption, there exists R such that if _V > R, then

{ } 〈z∗,u〉 ∗ inf ∥u∥V : z ∈ Au > M

and so for every z^∗∈ Au,

〈z∗,u〉-> M, 〈z∗,u〉 > M ∥u∥ ∥u∥V V

Thus if

_V > R,

{ } 〈z∗,u〉+-ε∥Lu-∥2V′ ∗ M--∥u∥V +-ε-∥Lu-∥2V′ inf ∥u∥V + ∥Lu ∥V′ : z ∈ Au ≥ ∥u∥V + ∥Lu∥V′

I claim that if

X is large enough, the above is larger than M∕2. If not, then there exists such that X →∞ but the right side is less than M∕2. First say is bounded. then there is an obvious contradiction since the right hand side then converges to M. Thus it can be assumed that V ′ →∞. Hence, for all n large enough, εV ′2 > MV ′. However, this implies the right side is larger than

M-∥un∥V-+-M-∥Lun-∥V′= M > M ∕2 ∥un∥V + ∥Lun∥V ′

This is a contradiction. Hence the right side is larger than M∕2 for all n large enough. It follows since M is arbitrary, that

{ } 〈z∗,u〉+ ε∥Lu ∥2V′ ∗ ∥ul∥im→∞ inf -∥u∥-+-∥Lu-∥-′-: z ∈ Au = ∞ X V V

It follows from Theorem 23.5.4 that if f ∈ V ^′, there exists u_ε such that for all v ∈ D

= X,

〈 −1 〉 ∗ ∗ ε Lv,F (Luε) V′,V + 〈Lu ε,v〉V′,V + 〈wε,v〉V′,V = 〈f,v〉, wε ∈ Auε (23.8.92)

(23.8.92)

First we get an estimate.

〈 −1 〉 ∗ ε Lu ε,F (Luε) V′,V + 〈Lu ε,uε〉V′,V + 〈wε,uε〉V ′,V = 〈f,uε〉

ε∥Luε∥2′ + 〈Luε,uε〉 ′ + 〈w∗,uε〉 ′ = 〈f,uε〉 V V ,V ε V ,V