Existence and uniqueness results for
variational inequalities
Lecture 1
Basic results
Didier Aussel
Univ. de Perpignan
– p. 1
Outline of lecture 1
I- Why V.I.?
II- Set-valued or not?
a- The monotone case
b- The quasimonotone case
II- First existence results
a- The linear case
b- The finite dimensional case
– p. 2
Notations
X a Banach space
X ∗ its topological dual (w∗ -top.)
h·, ·i the duality product
– p. 3
Notations
X a Banach space
X ∗ its topological dual (w∗ -top.)
h·, ·i the duality product
Stampacchia variational inequality (strong):
X∗
Let T : X → 2 be a map and C be a nonempty subset of X.
Find x̄ ∈ C such that there exists x̄∗ ∈ T (x̄) for which
hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
Notation : S(T, C) set of solutions (⊂ C).
– p. 3
I
Why variational inequalities?
– p. 4
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
(P )
find x̄ ∈ C : f (x̄) = inf f (x)
x∈C
– p. 5
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
(P )
find x̄ ∈ C : f (x̄) = inf f (x)
x∈C
Necessary condition: f is lsc + ...
if x̄ is a solution of (P ) then
x̄ ∈ S(∂f (x̄), C).
– p. 5
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
find x̄ ∈ C : f (x̄) = inf f (x)
(P )
x∈C
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
C a nonempty convex subset of X, x̄ ∈ C + C.Q.
Then
f (x̄) = inf f (x)
x∈C
⇐⇒
x̄ ∈ S(∂f, C)
– p. 5
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
find x̄ ∈ C : f (x̄) = inf f (x)
(P )
x∈C
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
C a nonempty convex subset of X, x̄ ∈ C + C.Q.
Then
f (x̄) = inf f (x)
x∈C
⇐⇒
x̄ ∈ S(∂f, C)
What about quasiconvex case? see the third lecture
– p. 5
Another motivation
Signorini’s frictionless contact problem
surface traction
f2 ∈ [L2 (Γ2 )]k
Γ1
volume force
f0 ∈ [L2 (Ω)]k
Γ2
ω
Γ3
@
I
@
@
@
@
@
Frictionless contact
– p. 6
Notations
Functional spaces:
Sk = {σ = (σij )ij ∈ Rk×k : σij = σji } = Rk×k
s
W = {v ∈ H 1 (Ω)k : v = 0 sur Γ1 }
Q = {q = (qij ) ∈ L2 (Ω)k×k : qij = qji , 1 ≤ i, j ≤ k} = L2 (Ω)k×k
s
W2 = {v ∈ W : vν ≤ 0 a.e. on Γ3 }
Deformation operator: ε : H 1 (Ω)k → Q
1
εij (u) =
2
Scalar product: hp, qiQ =
R
Ω
»
–
∂uj
∂ui
+
,
∂xj
∂xi
1 ≤ i, j ≤ k.
pij (x)qij (x) dx et hu, viW = hε(u), ε(v)iQ
Elasticity operator: F : Ω × Sk → Sk
Stress function : σ : H 1 (Ω)k → Q defined by
σ(u) :
Ω
→
Sk
x
7→
F(x, ε(u)(x)).
– p. 7
Formulations of the problem:
Find a displacement field u : Ω → R such that
−Div σ(u) = f0
on
Ω
u=0
on
Γ1
σ(u)ν = f2
on
Γ2
uν ≤ 0, σ(u)ν ≤ 0, σ(u)ν uν = 0, σ(u)τ = 0
on
Γ3
Variational formulation:
Find u ∈ W2 such that
hσ(u), ε(v) − ε(u)iQ ≥ hf, v − uiW ,
∀ v ∈ W2
where f is an element of W defined by
hf, viW =
Z
Ω
f0 .v dx +
Z
f2 .v da.
Γ2
– p. 8
II - Set-valued or not?
Stampacchia variational inequality (strong):
∗
Let T : X → 2X be a map and C be a nonempty subset of X.
Find x̄ ∈ C such that there exists x̄∗ ∈ T (x̄) for which
hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
– p. 9
II - Set-valued or not?
Stampacchia variational inequality (strong):
∗
Let T : X → 2X be a map and C be a nonempty subset of X.
Find x̄ ∈ C such that there exists x̄∗ ∈ T (x̄) for which
hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
Two particular cases:
a- The monotone case
b- The quasimonotone case
– p. 9
II - a
The monotone case
∗
Let T : X → 2X be a map
T is monotone iff ∀ x, y ∈ X, ∀ x∗ ∈ T (x) and ∀ y ∗ ∈ T (x)
hy ∗ − x∗ , y − xi ≥ 0.
– p. 10
Dense single-valuedness results
Theorem 1 (Zarantonello 1973)
X∗
Let X be a separable Banach space and let T : X → 2 be a
monotone operator. Then
the set of all x ∈ dom (T ) for which T (x) is not single-valued
has empty interior;
if X is finite-dimensional then this set has Lebesgue measure
zero.
– p. 11
Dense single-valuedness results
Theorem 2 (M. Lassonde ’09)
X∗
Let T : Z → 2 be minimal w -cusco from a Baire space Z into the
dual of a Banach space X. If there is a subset C ⊂ X ∗ with the w∗
-RNP such that the set {z ∈ Z : T (z) ∩ C 6= ∅} is dense in Z, then
there is a dense Gδ subset D of Z such that, at each point of D, T is
single-valued and upper semicontinuous.
∗
Maximal monotone operators T : Z → 2X with nonempty values, are typical examples of
minimal w -cusco mappings
A nonempty bounded subset A of the dual space X ∗ is said to be w∗ -dentable provided for every
ε > 0 there exists a w∗ -open half-space V in X ∗ such that A ∩ V 6= ∅ and diam(A ∩ V ) < ε.
A subset A of X ∗ is said to have the w∗ Radon-Nikodim Property provided every nonempty
bounded subset of A is w∗ -dentable.
– p. 12
Pointwise Single-valuedness
Theorem 3 (Kenderov 1975)
X∗
Let X be a Banach space and T : X → 2 be a monotone
set-valued map.
If T is lower semicontinuous at a point x0 ,
then T is single-valued at x0 .
T is lsc at x0 if, for every open V such that V ∩ T (x0 ) 6= ∅, there exists a
neigh. U of x0 such that V ∩ T (x) 6= ∅, for any x ∈ U .
– p. 13
Local Single-valuedness
Theorem 4 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
– p. 14
Local Single-valuedness
Theorem 5 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
T : X → 2Y is Lipschitz-like around (x, y) ∈ gr T if it exist a neighb. U of x, a
neighb. V of y and l > 0 such that
T (u) ∩ V ⊂ T (u′ ) + lku′ − ukB Y (0, 1),
∀ u, u′ ∈ U
– p. 14
Local Single-valuedness
Theorem 6 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
T : X → 2Y is Lipschitz-like around (x, y) ∈ gr T if it exist a neighb. U of x, a
neighb. V of y and l > 0 such that
T (u) ∩ V ⊂ T (u′ ) + lku′ − ukB Y (0, 1),
∀ u, u′ ∈ U
also called Aubin property or pseudo-Lipschitzianity.
– p. 14
Local Single-valuedness
Theorem 7 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
T : X → 2Y is Lipschitz-like around (x, y) ∈ gr T if it exist a neighb. U of x, a
neighb. V of y and l > 0 such that
T (u) ∩ V ⊂ T (u′ ) + lku′ − ukB Y (0, 1),
∀ u, u′ ∈ U
⇒ T is nonempty valued on U .
– p. 14
Local Single-valuedness
Theorem 8 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
T : X → 2Y is Lipschitz-like around (x, y) ∈ gr T if it exist a neighb. U of x, a
neighb. V of y and l > 0 such that
T (u) ∩ V ⊂ T (u′ ) + lku′ − ukB Y (0, 1),
∀ u, u′ ∈ U
If T is single-valued
T is Lipschitz-like around (x, T (x)) ⇔ T is loc. Lipschitz at x
– p. 14
II - b
The quasimonotone case
– p. 15
II - b
The quasimonotone case
or What about the quasimonotone case?
– p. 15
Quasimonotonicity
X∗
Let T : X → 2
be a set-valued map
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
– p. 16
Quasimonotonicity
X∗
Let T : X → 2
be a set-valued map
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
Examples:
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all x, y ∈ K and all t ∈ [0, 1]
f (tx + (1 − t)y) ≤ max{f (x), f (y)}.
– p. 16
Quasimonotonicity
X∗
Let T : X → 2
be a set-valued map
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
Examples:
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
– p. 16
Quasimonotonicity
X∗
Let T : X → 2
be a set-valued map
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
Examples:
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f differentiable
f is quasiconvex iff f ′ is quasimonotone
– p. 16
Quasimonotonicity
X∗
Let T : X → 2
be a set-valued map
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
Examples:
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f is quasiconvex iff ∂f is quasimonotone
– p. 16
Quasimonotonicity
X∗
Let T : X → 2
be a set-valued map
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
Examples:
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f is quasiconvex iff ∂f is quasimonotone
∗
the normal operator Nf : X → 2X defined by
x 7→ Nf (x) = N (Sf (x) , x)
is quasimonotone.
– p. 16
Links with other monotonicities
T is monotone iff ∀ x, y ∈ X, ∀ x∗ ∈ T (x) and ∀ y ∗ ∈ T (x)
hy ∗ − x∗ , y − xi ≥ 0.
T is pseudomonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi ≥ 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
monotone
⇓
pseudomonotone
⇓
quasimonotone
– p. 17
No hope in the quasimonotone case
No similar result (single-valued property) can be obtained in the
quasimonotone case!!.
Consider T : R → 2R defined by
T (x) = R++ .
then
T is pseudomonotone
T is Lipschitz-like at each point of its graph
(U , V any neigh., l any positive real)
T is multivalued!!!!
– p. 18
X∗
Definition 9 A set-valued map T : K → 2
is said to be
- single-directional at x ∈ dom T if, T (x) ⊆ R+ {x∗ } for some
x∗ ∈ T (x).
- locally single-directional at x ∈ dom T if it exists a neigh. U of
x such that, for any x′ ∈ U , T (x′ ) is single-directional at x′ .
- strictly single-directional at x if T (x) ⊆]0, +∞[{x∗ } for some
x∗ 6= 0 .
- locally strictly single-directional at x if T is strictly
single-directional at any point of a neigh. of x
– p. 19
Quasimonotone lsc case
X∗
Proposition 10 Let T : X → 2
semicontinuous at x ∈ X. Then
be a set-valued map, lower
i) if T is quasimonotone, then T is single-directional at x;
ii) if T is pseudomonotone, then T is strictly single-directional or
trivial at x.
– p. 20
Recovering Kenderov’73 (monotone lsc)
Proposition 11 (D.A., J.-N. Corvellec & M. Lassonde ’94)
X∗
A map T : X → 2 is monotone if and only if, for any α∗ ∈ X ∗ ,
the map T + {α∗ } is quasimonotone
– p. 21
Recovering Kenderov’73 (monotone lsc)
Proposition 13 (D.A., J.-N. Corvellec & M. Lassonde ’94)
X∗
A map T : X → 2 is monotone if and only if, for any α∗ ∈ X ∗ ,
the map T + {α∗ } is quasimonotone
X∗
Proposition 14 Let T : X → 2 be a set-valued map and x be a
point of its domain dom T be such that, for any α∗ ∈ X ∗ , the map
T + α∗ is single-directional at x, then T is single-valued at x.
– p. 21
Recovering Kenderov’73 (monotone lsc)
Hyp : T is monotone and lsc at x.
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone lsc at x
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is single-directional at x
⇒ T is single-valued at x.
– p. 22
Recovering Kenderov’73 (monotone lsc)
Hyp : T is monotone and lsc at x.
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone lsc at x
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is single-directional at x
⇒ T is single-valued at x.
Theorem 16 (Kenderov 1975)
X∗
Let X be a Banach space and T : X → 2 be a monotone
set-valued map.
If T is lower semicontinuous at a point x0 ,
then T is single-valued at x0 .
– p. 22