PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON
DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS
May 24 – 27, 2002, Wilmington, NC, USA
pp. 1-10
PRINCIPLE OF SYMMETRIC CRITICALITY
AND EVOLUTION EQUATIONS
Goro Akagi, Jun Kobayashi, and Mitsuharu Ôtani
Department of Applied Physics
School of Science and Engineering
Waseda University
3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan
Abstract. Let X be a Banach space on which a symmetry group G linearly acts
and let J be a G-invariant functional defined on X. In 1979, R. Palais [6] gave some
sufficient conditions to guarantee the so-called “Principle of Symmetric Criticality”:
every critical point of J restricted on the subspace of symmetric points becomes also
a critical point of J on the whole space X. In [5], this principle was generalized to
the case where J is non-smooth and the setting does not require the full variational
structure when G is compact or isometric.
The purpose of this paper is to combine this result with the abstract theory
developed in [1] and [2] concerning the evolution equation: du(t)/dt + ∂ϕ1 (u(t)) −
∂ϕ2 (u(t)) 3 f (t) in V ∗ , where ∂ϕi is the so-called subdifferential operator from a
Banach space V into its dual V ∗ . It is assumed that there exists a Hilbert space
H satisfying V ⊂ H ⊂ V ∗ and that G acts on these spaces as isometries. In this
setting, the existence of G-symmetric solution for above equation can be discussed.
As an application, a parabolic problem with the p-Laplacian in unbounded domains is discussed.
1. Introduction. Let X be a Banach space on which a group G linearly acts and
let J be a G-invariant (i.e., J(gu) = J(u) ∀ g ∈ G, ∀ u ∈ X) C 1 -functional on X.
Let ΣX be the subspace consisting of all symmetric points in terms of G, that is,
ΣX = {u ∈ X : gu = u ∀ g ∈ G}. Then the (classical) “principle of symmetric
criticality” reads
(P0 )
(J|ΣX )0 (u) = 0 =⇒ J 0 (u) = 0,
where J|ΣX is the restriction of J to ΣX . In 1979, R. S. Palais [6] studied this
principle and gave several sufficient conditions which assure the validity of it. In
particular, he showed that this principle is valid if G is a compact Lie group or if
X is a Hilbert space and G is isometric (actually he discussed in a more general
G-manifold setting). In his theory, however, one needs to work in the C 1 -category
with full variational structure.
In our earlier work [5], the classical principle (P0 ) was extended to the case where
the functional to be considered need not to be differentiable; let J be as above and
1991 Mathematics Subject Classification. Primary: 58E05, 58E40, 34G25; Secondary: 35K55.
Key words and phrases. symmetric criticality, group action, evolution equations, subdifferential, p-Laplacian, unbounded domain.
G. Akagi is supported by Waseda University Grant for Special Research Projects, #2002A-900.
J. Kobayashi is supported by Waseda University Grant for Special Research Projects, #2002A089. M. Ôtani is supported by the Grant-in-Aid for Scientific Research, #12440051, the Ministry
of Education, Culture, Sports and Technology, Japan and Waseda University Grant for Special
Research Projects, #2001B-017.
1
2
G. AKAGI, J. KOBAYASHI, AND M. ÔTANI
let ϕ be a G-invariant lower semicontinuous convex function from X into (−∞, +∞].
Then the “principle” (for subdifferentials) reads
(P1 )0
∂(ϕ|ΣX )(u) + (J|ΣX )0 (u) 3 0 =⇒ ∂ϕ(u) + J 0 (u) 3 0,
where ∂ϕ(u) denotes the subdifferential (a generalization of the Fréchet derivative)
of ϕ at u and becomes a multivalued operator in general. Roughly speaking, (P1 )0
says that every critical point of ϕ|ΣX + J|ΣX becomes also a critical point of ϕ + J.
Hence (P1 )0 can be regarded as an extension of the classical principle (P0 ). Actually,
in [5], much more “generalized principle” was introduced (see (P1 ) in §2.2) and
(P1 )0 is given as the special case which has full variational structure. It was shown
that such “generalized principle” is valid if G is compact or if X and its dual X ∗
are both reflexive and strictly convex and G is isometric. As an application of this
abstract result to non-smooth variational problems, an elliptic variational inequality
associated with the p-Laplacian in unbounded domains was treated.
The main objective of this paper is to exemplify the applicability of the abstract
setting of the “generalized principle” (P1 ) to evolution equations, which do not have
full variational structure. We shall relate this principle (P1 ) to abstract results in
[1] and [2] concerning evolution equations governed by subdifferentials in reflexive
Banach spaces. More precisely, let V be a reflexive Banach space on which G acts as
isometries and let ϕ1 and ϕ2 be G-invariant lower semicontinuous convex functions
from V into (−∞, +∞]. We shall study the existence of G-symmetric solutions of
the following Cauchy problem:
du
(CP)
(t) + ∂ϕ1 (u(t)) − ∂ϕ2 (u(t)) 3 f (t) in V ∗ , u(0) = u0 .
dt
According to [1] and [2], it is assumed that there exists a Hilbert space H satisfying
V ⊂ H ≡ H ∗ ⊂ V ∗ with densely defined continuous injections. Moreover we
shall assume that G also acts on H as isometries and the action on V coincides
with the restriction of the action on H. Then a suitable isometric group action on
X = L2 (0, T ; V ) is induced. We regard (CP) as an equation in X ∗ = L2 (0, T ; V ∗ )
and show that the time derivative becomes a G-equivariant operator (see §2.1) from
X into X ∗ as well as the (generalized) derivative of a G-invariant functional.
We apply our abstract result to a parabolic problem associated with the pLaplacian as in [2], where the domain is assumed to be bounded. We here discuss
the case where the domain is unbounded with some symmetry.
2. Principle of symmetric criticality. In this section we recall the results of
[5]. We begin with some definitions.
2.1. Group action. Let X be a real Banach space and let G be a group. We say
that X is a Banach G-space if G acts on X as a transformation group of bounded
linear operators in X, that is, a bounded linear operator πX (g) in X is assigned to
each g ∈ G and satisfies
πX (e)u = u ∀ u ∈ X,
πX (g1 g2 )u = πX (g1 )(πX (g2 )u)
∀
g1 , g2 ∈ G, ∀ u ∈ X,
where e is the identity element of G. πX is called a representation of G over X. Note
that G is not assumed to be a topological group nor the mapping [g, u] 7→ πX (g)u
assumed to be continuous. The dual space X ∗ of X becomes also a Banach G-space
by the relation:
hπX ∗ (g)v ∗ , ui = hv ∗ , πX (g −1 )ui g ∈ G, v ∗ ∈ X ∗ , u ∈ X.
(1)
PRINCIPLE OF SYMMETRIC CRITICALITY AND EVOLUTION EQUATIONS
3
We shall often write gu or gv ∗ instead of πX (g)u or πX ∗ (g)v ∗ respectively if no
confusion arises.
Let X be a Banach G-space. A function f on X is called G-invariant if
∀
f (gu) = f (u)
u ∈ X, ∀ g ∈ G
and a subset M of X is called G-invariant if
gM = {gu : u ∈ M } ⊂ M
∀
g ∈ G.
A (multivalued) mapping T from X into an another Banach G-space Y is called
G-equivariant if the domain D(T ) of T is a G-invariant subset of X and
T (πX (g)u) = πY (g)T (u)
∀
u ∈ X, ∀ g ∈ G.
The linear subspaces of G-symmetric points of X is defined as the common fixed
points of G:
ΣX = {u ∈ X : gu = u ∀ g ∈ G}.
It is clear that ΣX forms a closed linear subspace of X, so ΣX is regarded as a
Banach space with its induced topology.
2.2. The generalized “Principle”. Let X be a Banach G-space. Let Φ(X) be the
set of all proper lower semicontinuous convex functions ϕ from X into (−∞, +∞],
where “proper” means the effective domain D(ϕ) = {u ∈ X| ϕ(u) < +∞} of ϕ is
not empty. For u ∈ D(ϕ), the subdifferential ∂X ϕ(u) of ϕ at u is defined by
∂X ϕ(u) = {v ∗ ∈ X ∗ : ϕ(w) − ϕ(u) ≥ hv ∗ , w − ui ∀ w ∈ X}.
Then, as is well known, ∂X ϕ is a maximal monotone operator from X into X ∗ . For
simplicity of notation, we shall often write ∂ϕ instead of ∂X ϕ.
Let ΦG (X) be the set of all G-invariant functionals belonging to Φ(X), and let
ΓG (X ∗ ) be the set of all G-invariant weakly∗ closed convex subsets of X ∗ . In [5],
as a generalization of the classical “principle of symmetric criticality”, the following
principle (P1 ) is introduced.
(P1 ) For all ϕ ∈ ΦG (X) and all K ∈ ΓG (X ∗ ), it holds that
∂(ϕ|ΣX )(u) ∩ K|ΣX 6= ∅ =⇒ ∂ϕ(u) ∩ K 6= ∅,
where K|ΣX = {v ∗ |ΣX : v ∗ ∈ K} with (ΣX )∗ hv ∗ |ΣX , uiΣX :=
ing this principle, the following result holds.
X ∗hv
∗
, uiX . Concern-
Theorem 1. Assume that X is reflexive and the norms of X and X ∗ are both
strictly convex. Moreover, assume that the action of G is isometric, i.e.,
kguk = kuk
∀
u ∈ X, ∀ g ∈ G.
Then the principle (P1 ) is valid.
For later use, we prove a corollary of this theorem.
Corollary 1. Suppose that all the assumptions in Theorem 1 are satisfied. Let ϕ,
ψ ∈ ΦG (X), and f ∗ ∈ ΣX ∗ . Let M be a G-equivariant (single-valued) mapping
from D(M ) ⊂ X into X ∗ . Then it holds that
(P1 )00
M u + ∂(ϕ + IΣX )(u) − ∂ψ(u) 3 f ∗ ⇒ M u + ∂ϕ(u) − ∂ψ(u) 3 f ∗ ,
where IΣX is the indicator function of ΣX , i.e., IΣX (u) = 0 if u ∈ ΣX ; IΣX = +∞
if u \ ΣX .
4
G. AKAGI, J. KOBAYASHI, AND M. ÔTANI
Proof. Assume that
M u + ∂(ϕ + IΣX )(u) − ∂ψ(u) 3 f ∗
and put K := −M u + ∂ψ(u) − f ∗ . Then the above equation implies
∂(ϕ + IΣX )(u) ∩ K 6= ∅.
(2)
∂(ϕ|ΣX )(u) ∩ K|Σ 6= ∅.
(3)
We first prove
Indeed, by (2), there exists a v ∗ ∈ K such that v ∗ ∈ ∂(ϕ + IΣX )(u). This implies
u ∈ ΣX ∩ D(ϕ)
and ϕ(w) − ϕ(u) ≥ X ∗hv ∗ , w − uiX ∀ w ∈ ΣX ,
or
u ∈ D(ϕ|ΣX ) and
ϕ|ΣX (w) − ϕ|ΣX (u) ≥ (ΣX )∗hv ∗ |ΣX , w − uiΣX ∀ w ∈ ΣX .
Therefore we have v ∗ |ΣX ∈ ∂(ϕ|ΣX )(u). Thus (3) is verified.
We next show that K ∈ ΓG (X ∗ ). Since ∂ψ(u) is a convex closed subset of X ∗ , so
is K. Hence it suffices to show K is G-invariant. To do this, note that ∂ψ becomes
also G-equivariant by the G-invariance of ψ ([5, Proposition 3.1]). Therefore, since
u ∈ ΣX , we have
gM u = M (gu) = M u,
g∂ψ(u) = ∂ψ(gu) = ∂ψ(u)
∀
g ∈ G,
that is, M u ∈ ΣX ∗ and ∂ψ(u) is G-invariant. Thus K = −M u + ∂ψ(u) − f ∗ is also
G-invariant.
Finally, by Theorem 1, it follows from (3) that ∂ϕ(u) ∩ K 6= ∅, that is, M u +
∂ϕ(u) − ∂ψ(u) 3 f ∗ .
3. Evolution equations in Banach G-spaces. In this section, we combine the
result in the previous section with the abstract results of [2]. We study the evolution
equations governed by subdifferentials in reflexive Banach G-spaces.
3.1. Formulation. Let V be a real reflexive Banach space such that V and its
dual V ∗ are both strictly convex. We assume that there exists a real Hilbert space
H identified with its dual H ∗ such that
V ⊂ H ≡ H ∗ ⊂ V ∗,
(4)
where each injection is dense and continuous. Then it holds that
hu, viV = (u, v)H
∀u ∈ H, ∀v ∈ V.
(5)
Here and henceforth, we denote by h·, ·iV the duality pairing between V and V ∗
and by (·, ·)H the inner product of H respectively. We assume that a group G acts
on these spaces as isometries in the following manner:
(H.0) The action of G on V and H are both isometric and satisfy
πH (g)u = πV (g)u ∀g ∈ G, ∀u ∈ V.
By the assumption that the action of G on H is isometric, we have
1
1
(πH (g)v, u)H =
|πH (g)v + u|2H − |πH (g)v − u|2H
4
4
1
1
|v + πH (g −1 )u|2H − |v − πH (g −1 )u|2H
=
4
4
= (v, πH (g −1 )u)H
(6)
(7)
PRINCIPLE OF SYMMETRIC CRITICALITY AND EVOLUTION EQUATIONS
5
for all u, v ∈ H, and g ∈ H. This implies πH ∗ = πH and hence H ∗ can be identified
with H as a Banach G-space.
Moreover the action on H coincides with the restriction of the action on V ∗ :
Proposition 1. Assume that (H.0) is satisfied. Then the following holds:
πV ∗ (g)u = πH (g)u
∀g ∈ G, ∀u ∈ H.
(8)
Proof. Let g ∈ G and u ∈ H. Then it follows from (5), (6), and (7) that
hπV ∗ (g)u, viV
=
hu, πV (g −1 )viV
=
(u, πH (g −1 )v)H
=
=
(πH (g)u, v)H
hπH (g)u, viV .
for all v ∈ V , which implies (8)
By (6) and (8), we have ΣV ⊂ ΣH ⊂ ΣV ∗ . A V ∗ -valued function u on (0, T ) is
said to be G-symmetric if u(t) ∈ ΣV ∗ a.e. t ∈ (0, T ).
3.2. Main results. Let ϕ1 , ϕ2 ∈ ΦG (V ) be such that D(ϕ1 ) ∩ D(ϕ2 ) 6= ∅. We
study the G-invariant strong solution of the following Cauchy problem:
(CP)
du
(t) + ∂ϕ1 (u(t)) − ∂ϕ2 (u(t)) 3 f (t) in V ∗ ,
dt
u(0) = u0 .
Definition 1. A function u ∈ C([0, T ]; V ∗ ) is said to be a strong solution of (CP)
on [0, T ] if the following conditions are satisfied:
(i) u(·) is a V ∗ -valued absolutely continuous function on (0, T ],
(ii) u(0+) = u0 ,
(iii) u(t) ∈ D(∂ϕ1 ) ∩ D(∂ϕ2 ) for a.e. t ∈ (0, T ) and there exist sections ξ 1 (t) ∈
∂ϕ1 (u(t)) and ξ 2 (t) ∈ ∂ϕ2 (u(t)) satisfying:
du
(t) + ξ 1 (t) − ξ 2 (t) = f (t) in V ∗ a.e. on (0, T ).
dt
(9)
We here introduce the following four conditions, which are the analogues of (A.1)
– (A.3) in [2].
(H.1) kukrV − C1 |u|2H − C2 ≤ C3 ϕ1 (u) ∀u ∈ ΣV ,
(H.2) D(ϕ1 ) ∩ ΣV ⊂ D(∂ϕ2 ) and the following holds:
Let (un ) be a G-symmetric sequence of L∞ (0, T ; V ) ∩ W 1,2 (0, T ; H) such that
Z T
1
|dun (t)/dt|2H dt < +∞
sup {ϕ (un (t)) + |un (t)|H } < +∞ sup
n∈N
t∈[0,T ],n∈N
0
2
and let ξn (·) ∈ ∂ϕ ((un (·)). Then (ξn ) has a convergent subsequence in
C([0, T ]; V ∗ ).
(H.3) There exists ϕ̃2 ∈ Φ(H) such that ϕ̃2 (u) = ϕ2 (u) for all u ∈ V , Jλ2 (D(ϕ1 ) ∩
ΣV ) ⊂ D(ϕ1 ) ∩ ΣV , and
¡
¢
ϕ1 (Jλ2 u) ≤ l1 ϕ1 (u) + l2 (|u|H ) ∀u ∈ D(ϕ1 ) ∩ ΣV , ∀λ > 0,
where C1 , . . . , C5 are positive constants, l1 , l2 are positive monotone increasing
functions on [0, +∞), r ∈ (1, +∞), and Jλ2 denotes the resolvent of ∂H ϕ̃2 , i.e.,
Jλ2 = (I + λ∂H ϕ̃2 )−1 .
6
G. AKAGI, J. KOBAYASHI, AND M. ÔTANI
Theorem 2. Assume that (H.0) – (H.3) are satisfied. Then for all u0 ∈ D(ϕ1 )∩ΣV
0
and any G-symmetric function f ∈ W 1,r (0, T ; V ∗ ) (r0 = r/(r − 1)), there exists a
number T0 ∈ (0, T ] such that (CP) has a strong solution u on [0, T0 ] satisfying:
u ∈ Cw ([0, T0 ]; V ) ∩ W 1,2 (0, T0 ; H),
u(t) ∈ D(∂ϕ1 ) ∩ D(∂ϕ2 ) ∩ ΣV for a.e. t ∈ (0, T0 ),
ξ 1 ∈ L2 (0, T0 ; V ∗ ),
ξ 2 ∈ C([0, T0 ]; V ∗ ),
1
sup ϕ (u(t)) < +∞,
ϕ2 (u(·)) ∈ C([0, T0 ]),
t∈[0,T0 ]









(10)
where Cw ([0, T0 ]; V ) denotes the set of all V -valued weakly continuous functions on
[0, T0 ], and ξ 1 and ξ 2 are sections of ∂ϕ1 and ∂ϕ2 satisfying (9) with T replaced by
T0 .
3.3. Proof of Theorem 2. Consider the following auxiliary equation:
du
(t) + ∂(ϕ1 + IΣV )(u(t)) − ∂ϕ2 (u(t)) 3 f (t) in V ∗ ,
dt
u(0) = u0 .
(11)
Let ϕ1Σ := ϕ1 + IΣV . Then it holds that ϕ1Σ ∈ Φ(V ) and D(ϕ1Σ ) = D(ϕ1 ) ∩ Σ.
Therefore, by (H.1) – (H.3), the conditions (A.1) – (A.3) of [2] are satisfied with
ϕ1 replaced by ϕ1Σ . Hence, by Theorem 1 of [2], there exists a local in time strong
solution u on [0, T0 ] of (11) satisfying
u ∈ Cw ([0, T0 ]; V ) ∩ W 1,2 (0, T0 ; H),
supt∈[0,T0 ] ϕ1 (u(t)) < +∞,
ϕ2 (u(·)) ∈ C([0, T0 ])
¾
(12)
and there exists sections χ1 (t) ∈ ∂(ϕ1 + IΣV )(u(t)) and χ2 (t) ∈ ∂ϕ2 (u(t)) such that
χ1 ∈ L2 (0, T0 ; V ∗ ),
χ2 ∈ C([0, T0 ]; V ∗ ),
(13)
du
(t) + χ1 (t) − χ2 (t) = f (t) in V ∗ , a.e. t ∈ (0, T0 ).
dt
We are going to show that u turns out to be a strong solution of (CP) by applying
Corollary 1 and proving that (10) is satisfied. For simplicity, we assume that T0 = T .
Let us put X := L2 (0, T ; V ) and define the action of G on X by
(πX (g)v)(t) = πV (g)(v(t)) a.e. t ∈ (0, T ).
(14)
Then, since the action of G on V is isometric, this action also becomes isometric.
Proposition 2. As for the action of G on X ∗ (= L2 (0, T ; V ∗ )), we have:
(i) (πX ∗ (g)v)(t) = πV ∗ (g)(v(t)) a.e. t ∈ (0, T )
(ii) πX ∗ (g)v = πX (g)v ∀ v ∈ X, ∀ g ∈ G.
∀
g ∈ G, ∀ v ∈ X ∗ .
Proof of Proposition 2. Here and henceforth we denote by h·, ·iX the duality pairing
between X and X ∗ :
Z T
hv, wiX =
hv(t), w(t)iV dt v ∈ X ∗ , w ∈ X.
0
PRINCIPLE OF SYMMETRIC CRITICALITY AND EVOLUTION EQUATIONS
7
Let g ∈ G and v ∈ X ∗ . Then, by (1) and (14), we have
Z T
h{πX ∗ (g)v}(t), w(t)iV dt = hπX ∗ (g)v, wiX
0
= hv, πX (g −1 )wiX
Z T
=
hv(t), πV (g −1 ){w(t)}iV dt
Z
0
T
hπV ∗ (g){v(t)}, w(t)iV dt
=
0
for all w ∈ X. Therefore the assertion (i) holds. Furthermore, if v ∈ X, then (6),
(8), and (14) yield
(πX ∗ (g)v)(t) = πV ∗ (g)(v(t)) = πV (g)(v(t)) = (πX (g)v)(t) a.e. t ∈ (0, T ).
We next show the G-equivariance of the differential operator d/dt. Define the
mapping M from X into X ∗ by
dv
(t) a.e. t ∈ (0, T )
dt
with the domain D(M ) = {v ∈ X : dv/dt ∈ X ∗ }.
(M v)(t) =
Proposition 3. The mapping M defined above is G-equivariant from X into X ∗ .
Proof of Proposition 3. By the definition of the time derivative and the basic
property of Bochner integral (see e.g. [3, Definition 1.4.28 and Proposition 1.4.22]),
it holds that M v = w if and only if
Z T
Z T
(15)
η(t)hw(t), hiV dt = −
η 0 (t)hv(t), hiV dt ∀ h ∈ V, ∀ η ∈ C0∞ (0, T ).
0
0
Let v ∈ D(M ) and g ∈ G. By Proposition 2 and (15), we obtain
Z T
Z T
η(t)h{πX ∗ (g)M v}(t), hiV dt =
η(t)h(M v)(t), πV (g −1 )hiV dt
0
0
Z
=
T
−
η 0 (t)hv(t), πV (g −1 )hiV dt
0
Z
=
−
T
η 0 (t)h{πX (g)v}(t), hiV dt
0
for all h ∈ V and η ∈ C0∞ (0, T ). Therefore it follows from (15) that
M (πX (g)v) = πX ∗ (g)(M v),
that is, M is G-equivariant.
We are now in a position to apply Corollary 1. We first observe that (14) yields
Z T
IΣX (v) =
IΣV (v(t))dt.
0
We next define two lower semicontinuous convex functions Φ1 and Φ2 from X into
(−∞, +∞] by
Z T
Φi (v) =
ϕi (v(t))dt (i = 1, 2).
0
8
G. AKAGI, J. KOBAYASHI, AND M. ÔTANI
Then, for v ∈ X and v ∗ ∈ X ∗ , it holds that v ∗ ∈ ∂Φi (v) if and only if v ∗ (t) ∈
∂ϕi (v(t)) a.e. t ∈ (0, T ) ([4, Proposition 1.1]). Similarly, v ∗ ∈ ∂(Φ1 + IΣX )(v) if
and only if v ∗ (t) ∈ ∂(ϕ1 + IΣV )(v(t)) a.e. t ∈ (0, T ). Hence, using (12) and (13),
we obtain from (11) that
M u + ∂(Ψ1 + IΣX )(u) − ∂Ψ2 (u) 3 f
in X ∗ .
By (14), Φ1 and Φ2 become G-invariant, i.e., Φ1 , Φ2 ∈ ΦG (X). Therefore, applying
Corollary 1 with ϕ = Φ1 and ψ = Φ2 , we obtain
M u + ∂Ψ1 (u) − ∂Ψ2 (u) 3 f
in X ∗ .
By [4, Proposition 1.1] again, there exists ξ 1 , ξ 2 ∈ X ∗ = L2 (0, T ; V ∗ ) satisfying (iii)
of Definition 1. Thus we can conclude that u is a strong solution of (CP). Finally,
ξ 2 ∈ C([0, T ]; V ∗ ) follows from (H.2) with un = u and ξn = ξ.
4. Application. Let Ω be an (unbounded) domain in RN with smooth boundary
∂Ω and let G be a subgroup of O(N ) whose elements leave Ω invariant: g(Ω) = Ω
for all g ∈ G. We assume that Ω is compatible with G (see [7]), that is, for some
ρ>0
m(y, ρ, G) → ∞ as |y| → ∞ with dist(y, Ω) ≤ ρ,
(16)
where
½
¾
∃g1 , g2 , . . . , gn ∈ G s.t.
m(y, ρ, G) = sup n ∈ N :
B(gj y, ρ) ∩ B(gk y, ρ) = ∅ if j 6= k .
For u ∈ D = C0∞ (Ω) and g ∈ G, we define gu by
(gu)(x) = u(g −1 x) g ∈ G, x ∈ Ω.
(17)
0
For a distribution U (U ∈ D ) and g ∈ G, we also define gU by
hgU, ηi = hU, g −1 ηi
∀
η ∈ C0∞ (Ω).
We say U ∈ D0 is G-invariant if gU = U for all g ∈ G. It is easily seen that if
u ∈ L1loc (Ω), then gu is given by (17). Especially u ∈ L1loc (Ω) is G-invariant if and
only if u(gx) = u(x) for all g ∈ G and a.e. x ∈ Ω. Finally a function f from (0, T )
into D0 is said to be G-invariant if f (t) is G-invariant a.e. t ∈ (0, T ).
Throughout this chapter, we denote by C positive constants which do not depend
on the elements of the corresponding space or set. For p ∈ (1, +∞), p0 designates
the Hölder conjugate of p, i.e., p0 = p/(p − 1) and p∗ designates Sobolev’s critical
exponent, i.e., p∗ = N p/(N − p) (for p < N ); p∗ = +∞ (for N ≤ p).
Similarly to [2], we consider the following initial boundary value problem:

∂u


− ∆p u + µ|u|p−2 u − |u|q−2 u = f, in Ω × [0, T ],
∂t
(NHE)
u = 0
in ∂Ω × [0, T ],


u(x, 0) = u0 , x ∈ Ω,
where ∆p u := divx (|∇u|p−2 ∇u) is the so-called p-Laplacian and µ is a real parameter.
Theorem 3. Let 2 ≤ p and 2 < q < p∗ and let µ ≥ 0 if p = 2. Let f ∈
W 1,2 (0, T ; V ∗ ) and u0 ∈ V be G-invariant, where V := W01,p (Ω) ∩ L2 (Ω). Then
there exists a number T0 ∈ (0, T ] such that (NHE) has a G-invariant solution on
[0, T0 ] satisfying:
u ∈ Cw ([0, T0 ]; V ) ∩ C([0, T0 ]; Lq (Ω)) ∩ W 1,2 (0, T0 ; L2 (Ω)).
(18)
PRINCIPLE OF SYMMETRIC CRITICALITY AND EVOLUTION EQUATIONS
0
9
0
Theorem 4. Let p < 2, p < q < p∗ , and µ > 0. Let f ∈ W 1,p (0, T ; W −1,p (Ω))
and u0 ∈ W01,p (Ω) be G-invariant. Then there exists a number T0 ∈ (0, T ] such
that (NHE) has a G-invariant solution on [0, T0 ] satisfying (18) with V replaced by
W01,p (Ω).
Proof of Theorem 3. Set V = W01,p (Ω) ∩ L2 (Ω) and H = L2 (Ω) with norms
kukV := |∇u|p + |u|2 and |u|H = |u|2 respectively. Then V and V ∗ become reflexive
and strictly convex and V , H, and V ∗ satisfy (4). Define the action of G on V and
H by (17). Then (H.0) is also satisfied.
We first consider the case where µ = 0. Let ϕ1 and ϕ2 ∈ ΦG (V ) be given by
Z
Z
1
1
1
p
2
ϕ (u) =
|∇u(x)| dx, ϕ (u) =
|u(x)|q dx
p Ω
q Ω
Then ∂ϕ1 (u) and ∂ϕ2 (u) become −∆p u and |u|q−2 u in the sense of distribution,
where D(ϕ1 ) = D(∂ϕ1 ) = V and D(ϕ2 ) = D(∂ϕ2 ) = V respectively. (Note that ϕ1
and ϕ2 are Fréchet differentiable. So the subdifferentials coincide with their Fréchet
derivatives respectively.) Therefore (NHE) is reduced to the abstract equation (CP).
We next show that (H.1), (H.2), and (H.3) hold. We observe that
kuk2V
≤ 2|∇u|2Lp + 2|u|2L2
≤ 2(|∇u|pLp + 1) + 2|u|2L2
= 2pϕ1 (u) + 2 + 2|u|2H .
(19)
Hence (H.1) with r = 2 holds. In order to verify (H.2), we need the following
proposition, which is an analogue of Theorem 1.24 of [7].
Proposition 4. If Ω is compatible with G, then the embeddings
ΣV ⊂ Lq (Ω), 2 < q < p∗
are compact.
The proof of this proposition is given in [5, Proposition 4.2] and is quite similar
to that of Theorem 1.24 of [7], so we omit it.
Now let (un ) be a G-invariant sequence such that
¯
Z T¯
¯ dun ¯2
1
¯
¯
sup {ϕ (un (t)) + |un (t)|H } < C,
¯ dt (t)¯ dt < C.
t∈[0,T ]
0
H
Then, by (19), we have
sup ku(t)kV < C,
(20)
t∈[0,T ]
in particular (un (t)) is bounded in V for each t ∈ [0, T ]. Therefore, by Proposition
4, (un (t)) is compact in Lq (Ω) for all t ∈ [0, T ]. On the other hand, since 2 < q < p∗ ,
the Gagliardo-Nirenberg inequality and (20) yield
|un (t) − un (s)|Lq
≤ C|un (t) − un (s)|θH kun (t) − un (s)k1−θ
W 1,p
ÃZ ¯
!θ/2
¯
T ¯
¯2
¯ dun (t)¯ dt
≤ C|t − s|θ/2
¯
¯ dt
0
H
for some θ ∈ (0, 1). Therefore un (·) is equi-continuous in C([0, T ]; Lq (Ω)). Hence,
by Ascoli’s theorem, (un ) has a convergent subsequence (un0 ) in C([0, T ]; Lq (Ω)).
0
This implies |un0 |q−2 un0 → |u|q−2 u strongly in C([0, T ]; Lq (Ω)) for some u and thus
2
2
∗
∂ϕ (un0 ) → ∂ϕ (u) strongly in C([0, T ]; V ).
10
G. AKAGI, J. KOBAYASHI, AND M. ÔTANI
We are going to verify (H.3). Define ϕ̃2 by ϕ̃2 (u) = ϕ2 (u) if u ∈ Lq (Ω); ϕ̃2 (u) =
+∞ if u ∈ H \ Lq (Ω). Then we can easily see that ϕ2 = ϕ̃2 on V and ∂H ϕ̃2 (u) =
|u|q−2 u with D(∂H ϕ̃2 ) = L2 (Ω) ∩ L2(q−1) (Ω). For arbitrary u ∈ D(ϕ1 ) = V , take
un ∈ C0∞ (Ω) such that un → u in V as n → ∞. Let λ > 0 and vn = Jλ2 un , i.e.,
vn + λ|vn |q−2 vn = un . Put β(s) = |s|q−2 s. Then we have
vn (x) = (1 + λβ)−1 un (x)
∀
x∈Ω
(21)
Since (1 + λβ)−1 is in C 1 (R) and non-expansive, we see that vn ∈ C 1 (R) and
¯
¯ ¯
¯
¯ vn (x + h) − vn (x) ¯ ¯ un (x + h) − un (x) ¯ ∀
¯
¯≤¯
¯
x ∈ Ω, ∀ h ∈ RN with x + h ∈ Ω,
¯
¯ ¯
¯
h
h
whence it follows that |∇vn (x)| ≤ |∇un (x)| for all x ∈ Ω. Hence ϕ1 (vn ) ≤ ϕ1 (un ).
Since vn → v = Jλ2 u in H, we see that ϕ1 (Jλ2 u) ≤ lim inf ϕ1 (vn ). On the other
hand, we have lim ϕ1 (un ) = ϕ1 (u). Thus, we get ϕ1 (Jλ2 u) ≤ ϕ1 (u). Moreover if u is
G-invariant, then it easily follows from (21) with un , vn replaced by u, v that v is
also G-invariant, in other words, if u ∈ ΣV , then Jλ2 u ∈ ΣV . Thus (H.3) is satisfied
with l1 ≡ 1 and l2 ≡ 0.
Therefore, Theorem 2 assures the existence of a G-invariant local solution u of
(NHE). Moreover the regurality (18) follows from (10).
As for the case µ > 0, we put
Z
Z
1 T
1 T q
ϕ1 (u) =
(|∇u|p + µ|u|p )dx, ϕ2 (u) =
|u| dx
p 0
q 0
and for the case µ < 0, we put
Z
Z
Z
1 T
µ T p
1 T q
ϕ1 (u) =
|∇u|p dx, ϕ2 (u) = −
|u| dx +
|u| dx.
p 0
p 0
q 0
Then, by almost the same argument as above, we can ensure all the the required
assumptions as above. (Note that the embedding ΣV ⊂ Lp (Ω) is also compact when
p > 2).
Proof of Theorem 4. For this case, we set V = W01,p (Ω) and H = L2 (Ω) with usual
norms. Then we can repeat almost the same arguments as in the proof of Theorem
3 with obvious modifications (e.g., (H.1) should hold with r = p).
REFERENCES
[1] G. Akagi and M. Ôtani, Evolution inclusions governed by subdifferentials in reflexive Banach
spaces, preprint.
[2] G. Akagi and M. Ôtani, Evolution equations and subdifferentials in Banach spaces, to appear
in Proceedings of the fourth international conference on dynamical system and differential
equations, 2002, Wilmington.
[3] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon
Press, Oxford, 1998.
[4] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975),
304-331.
[5] J. Kobayashi and M. Ôtani, The principle of symmetric criticality for non-differentiable
mappings, preprint.
[6] R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys. 69 (1979), 19-30.
[7] M. Willem, “Minimax Theorem”, Birkäuser, Boston, 1996.
Received September 2002; in revised March 2003.
E-mail address: jun@otani.phys.waseda.ac.jp (J. Kobayashi)