An Estimate on Linear Functionals’ Kernels in Banach Spaces, and Regularity of Convex Functionals ()
1. Introduction
Many important differential equations are concerned with derivatives of convex functional defined on real Banach spaces.
This paper’s research is motivated to fined u in a Banach space
such that
(1.1)
where
denotes the Fréchet differential of a functional
and
is the subdifferential of a lower semi-continuous convex functional
. In general, for a proper convex functional
and
, the subgradients of
at u are the elements
satisfying
and the subdifferential
is the set of all subgradients of
at u (see [1]).
For a most interesting example, put
,
, and
(1.2)
If we can find u as a solution of (1.1) with (1.2), then there is
such that
is a critical point of
. In this example, to find the second critical point is very interesting since the mountain pass theorem is not so useful.
The author wants to verify the following assertion.
Assertion 1.1. Fix
,
. Assume that there are
satisfying
(1.3)
(1.4)
Then, for
,
and
are homeomorphic. Here,
and
means both
and
.
Assertion 1.1 is a kind of Morse lemma in the sense that the contraposition implies the existence of solutions of (1.1) in the case where
is compact and
(cf. [2] [3]). In trying to prove Assertion 1.1, the author obtained a number of propositions, and some of them seem to be useful in other mathematical researches. This paper’s theorems are obtained in such process.
2. Results
Assertion 1.1 is proved if we can define a Lipschitz continuous vector field
such that the flow
,
defines a homeomorphism between
and
. For example,
is expected to satisfy the following property.
(a) If
, then
.
(b) If
,
(b-1) if
, then
and
, or
(b-2) if
, then
, or
(b-3) if
, then
is continuous with (b-1) (b-2).
(c) If
,
is continuous with (a) (b).
In general, we cannot construct a Lipschitz continuous vector field
with (a)-(c). The author has constructed a sequence
such that each
is Lipschitz continuous with the constant
, and that
local uniformly. To do this, assumptions (1.3) (1.4) play an important role to see
(2.1)
Here, for
and
,
Hence, in the case where
satisfies
(2.2)
then for
and
the convergence
holds and
is the aimed homeomorphism with some
.
The author thinks, at this moment, that (2.2) can be hold if the following assumptions are satisfied.
(i)
is even, or equivalently
.
(ii)
such that
, for
,
.
(iii) (i) (ii) together mean that putting
for
defines a norm in the linear space
. Suppose that this norm is uniformly convex and uniformly smooth in Y.
(iv)
is compact in X.
In constructing the sequence
required above, the following theorem plays important roles, and seems to be useful in many other mathematical researches.
Theorem 2.1. Let X be a real Banach space, and
be the dual space of X. For
and
,
(2.3)
In the next two theorems, the differentiability and continuity of derivatives of convex functionals are shown.
For a moment, we recall definitions of derivatives (cf. Masuda [2]). Let
, where
are normed vector spaces and U is an open subset of V.
Definition 2.1. (Fréchet derivative) F is called Fréchet differentiable at
if there is a bounded linear operator
such that
Equivalently,the first-order expansion holds,in Landau notion
Definition 2.2. (Gâteaux derivative) The Gâteaux differential
of F at
in the direction
is defined as
If the limit exists for all
, then one calls F is Gâteaux differentiable at
.
The Gâteaux differential may fail to be linear, unlike the Fréchet derivative. Even if linear, it may fail to depend continuously on
.
In the following, let
be a lower semi-continuous convex functional. The set
is called the effective domain of
.
Remark 2.1. For
, put
Suppose that the Gâteaux differential
for every direction
exists. Then, since
is convex,
is a linear subspace of X and
is linear with respect to
.
Theorem 2.2. Let
. If U is open in X and
is Gâteaux differentiable at x, then
is Fréchet differentiable at
.
Remark 2.2. In Theorem 2.2, the openness of U is needed. For example, put
where C is a closed convex subset of X. Then
is a lower semi-continuous convex functional on X. As is noted in Remark 2.1, the Gâteaux differential
exists for all
and is linear on
, where
If 0 is not an inner point of C, or equivalently
, then
is not Fréchet differentiable at 0.
Theorem 2.3. Suppose
, U is open in X, and
is Fréchet differentiable on U. Then the Fréchet derivative of
is continuous on U.
3. Proof of Theorem 2.1
Throughout this paper, the following symbols are used.
For any
,
is expressed by
(3.1)
Let
. Take
such that
. Then, since
,
Noting the relation
in (3.1) implies
(3.2)
Since
is a linear subspace,
. Therefore,
On the other hand, the relation
implies
Thus, Theorem 2.1 is proved.
4. Proof of Theorem 2.2
Let
be an open subset of U satisfying
.
We verify that the linear functional
is bounded. For
, put
Since
is lower semi-continuous,
is closed in
. By Baire category theorem, the inclusion relationship
implies that, for some
,
has an inner point. Therefore,
is not dense in X. Hence also
. This means that
is closed in X, or equivalently,
is bounded (cf. Rudin [4]).
Now, put
Since
is lower semi-continuous convex functional on
satisfying
the Fréshet differentiability at
is proved if for each
there is
such that
(4.1)
To see this, fix any
. Put
Since
and
,
. Thus, Baire category theorem implies that
has an inner point
. Take
such that
.
If
, then taking
such that
implies (4.1). Hence, the proof is finished.
In the case where
, take the following closed cone.
Then, taking
instead of
in the same discussion implies that there is an open ball
. Since
is convex, the convex hull
is an open subset of
, and 0 is included. Thus, Theorem 2.2 is proved.
5. Proof of Theorem 2.3
Suppose that the result is not true. Then, there are
and a sequence
in U such that for some
For each k, there is
satisfying
. Hence, for all
,
(5.1)
where in the first inequality, the convexity of
is used.
On the other hand, since
is Fréchet differentiable at
,
Thus, (5.1) implies
(5.2)
Take
such that
Then, taking k such that
in (5.2) yields that
, which is a contradiction. Therefore, the aimed result is true.