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5.2 Abstract Existence Theorems for Variational Problems 225

Lemma 5.2.5. Let the sesquilinear form a(·, ·) satisfy G˚arding’s inequality
(5.2.2) and, in addition, have the property
Re a(v, v) > 0 for all v ∈H with v =0 .

Then a(·, ·) satisfies (5.2.4) and, consequently, is strongly H–elliptic.
Proof: The proof rests on the well known weak compactness of the unit
sphere in reflexive Banach spaces and in Hilbert spaces (Schechter [270, VIII
Theorem 4.2]), i. e. every bounded sequence {v j } j∈IN ⊂H with
v j
H ≤ M
contains a subsequence v j with a weak limit v 0 ∈H such that

lim (g, v j ) H =(g, v 0 ) H for every g ∈H .
j →∞
We now prove the lemma by contradiction. If a were not strongly H–
elliptic then there existed a sequence {v j } j∈IN ⊂H with
v j
H =1 and

lim Re a(v j ,v j )=0 .
j→∞
Then {v j } contained a subsequence {v j } converging weakly to v 0 ∈H.
G˚arding’s inequality then implied

2
α 0
v j − v 0
≤ Re{a(v j − v 0 ,v j − v 0 )+ C(v j − v 0 ),v j − v 0 }
H
H
≤ Re{a(v j ,v j ) − a(v 0 ,v j ) − a(v j ,v 0 )
+ a(v 0 ,v 0 )+(Cv j ,v j − v 0 ) H − (Cv 0 ,v j − v 0 ) H } .
Since C is compact, there existed a subsequence {v j }⊂{v j }⊂H such that

Cv j → w ∈H for j →∞. Hence, due to the weak convergence v j v 0
we would have

(Cv j ,v j ) H − (Cv j ,v 0 ) H → (w, v 0 ) H − (w, v 0 ) H =0 ,

∗ ∗
a(v 0 ,v j )= (jA) v 0 ,v j → (jA) v 0 ,v 0 = a(v 0 ,v 0 )
H H
and corresponding convergence of the remaining terms on the right–hand
side. This yielded

2
0 ≤ lim j →∞ α 0
v j − v 0
≤−Re a(v 0 ,v 0 )
H
and, consequently, with Re a(v 0 ,v 0 ) ≥ 0, we could find
lim
v j − v 0
H = 0 together with Re a(v 0 ,v 0 )=0 .
j →∞
The latter implied
v 0 = 0 ; however,
v 0
H = lim
v j
H =1 ,
j →∞