1. Introduction
Suppose $\Omega \subset \mathbb {R}^n$
is a bounded domain (open, connected) with boundary $\partial \Omega$
, and let $\Sigma \subset \partial \Omega$
be a subset of the boundary. If there exists a positive constant $C>0$
such that the inequality
with $d_\Sigma = \textrm {dist}(\cdot,\,\Sigma )$
is valid, we say that the Hardy inequality is satisfied for the pair $(\Omega,\,\Sigma )$
. Such inequalities are known to hold in a variety of settings, and for the particular cases $\Sigma = \{ \sigma \}$
(point singularities) and $\Sigma =\partial \Omega$
(the entire boundary) the relevant literature is quite extensive (especially for the latter case).
An important aspect of the theory of Hardy inequalities is to specify the best constant for a particular pair $(\Omega,\,\Sigma )$
, that is, the quantity
which is called the Hardy constant.
The particular case $\Sigma = \partial \Omega$
has been studied extensively and, apart from the well-known convexity condition, other conditions have been introduced which guarantee that the Hardy constant assumes the critical value $1/4$
, see [Reference Avkhadiev1, Reference Barbatis, Filippas and Tertikas3, Reference Davies9, Reference Gkikas13, Reference Marcus, Mizel and Pinchover16] and references therein.
The case of the point singularity has also been studied in recent years. Sufficient conditions to have $H(\Omega,\,\{ \sigma \})= n^2/4$
(again, the critical value) have been obtained in [Reference Barbatis, Filippas and Tertikas4, Reference Fall10], see also [Reference Cazacu7, Reference Chen and Véron8, Reference Fall and Musina12] for results within this context.
The intermediate case where $\Sigma$
is a part of the boundary which is larger than a single point and smaller than the entire boundary has been addressed in [Reference Fall and Mahmoudi11], in which the authors discuss the smooth case where $\partial \Omega$
is smooth and $\Sigma$
is a submanifold of $\partial \Omega$
of dimension $\dim \Sigma = s$
.
In this paper, we are primarily concerned with the behaviour of the Hardy constant under perturbations of the domain and the subset $\Sigma$
. In particular, if $\varphi$
is a diffeomorphism, we get a map
and our task is to investigate questions of continuity and differentiability of that map in an appropriate sense which is made precise in the next section. This problem has already been studied in a more general $L^p$
setting for the special case $\Sigma = \partial \Omega$
in [Reference Barbatis and Lamberti5], so our work here is a natural continuation of that work.
We also concern ourselves with the problem where only the submanifold is perturbed. This is expressed in a very neat way in the case of a point singularity: if we regard the Hardy constant as a function $H:\partial \Omega \rightarrow \mathbb {R}$
,
then this function is differentiable on $\partial \Omega$
, under some reasonable assumptions.
As a final remark, let us note that to extend this theory for $p\neq 2$
, results similar to those of [Reference Fall and Mahmoudi11] which relate the value of the best constant with the existence of minimizers (which we use in the sequel) would be required. This leaves an important open problem, which is in fact very interesting in its own right. Another issue is the regularity of the domain: in the results concerning differentiability of the Hardy constant, smoothness is assumed, since it is assumed in the results of [Reference Fall and Mahmoudi11], that are an important part of our approach. A conjecture is that these results can probably go as low as $C^{1,\alpha }$
in regularity, see for example [Reference Lamberti and Pinchover15] for similar results for the case $\Sigma = \partial \Omega$
.
2. Diffeomorphism groups
In this section we offer a quick review of finite-order diffeomorphism groups in $\mathbb {R}^n$
. For details, see [Reference Banyaga2]. A $C^k$
-diffeomorphism of $\mathbb {R}^n$
is a homeomorphism $\varphi : \mathbb {R}^n \rightarrow \mathbb {R}^n$
that is $k$
-times bi-differentiable. The set of all such maps is denoted by $Diff^k(\mathbb {R}^n)$
. It is obviously a group under composition. For our purposes, it is sufficient to work with the subgroup $Diff^k_c(\mathbb {R}^n)$
of $C^k$
-diffeomorphisms with compact support; the support of a diffeomorphism is defined to be the closure of the set of points that the diffeomorphism acts upon non-trivially, i.e.
Since we work on bounded domains, this is done without loss of generality, and spares us some technical considerations that are consequence of the non-compactness of $\mathbb {R}^n$
.
We now equip $Diff^k_c(\mathbb {R}^n)$
with the weak $C^k$
topology (or compact-open topology). To describe this topology, it suffices to describe the basic open sets that generate it. These are the ‘balls’:
of centre $\varphi \in Diff^k_c(\mathbb {R}^n)$
, radius $\epsilon >0$
and domain $K$
, which is a compact subset of $\mathbb {R}^n$
. Here, we assume
In this topology, $Diff^k_c(\mathbb {R}^n)$
is a topological group, which is in fact locally homeomorphic to the Banach space of $C^k$
vector fields of compact support $\mathfrak {X}^k_c(\mathbb {R}^n) \cong C^k_c(\mathbb {R}^n,\,\mathbb {R}^n)$
, thus assuming the structure of an infinite dimensional Lie group.
The directional derivative of a continuous function $H:Diff^k_c(\mathbb {R}^n) \rightarrow \mathbb {R}$
at $\varphi \in Diff^k_c(\mathbb {R}^n)$
in the direction of $\xi \in \mathfrak {X}^k_c(\mathbb {R}^n)$
is given by the limit
provided it exists. Note that the compact support assumption guarantees that $\varphi + t\xi$
is always a diffeomorphism provided that $t$
is small enough. If this is defined for all $\varphi \in Diff^k_c(\mathbb {R}^n)$
and all $\xi \in \mathfrak {X}^k_c(\mathbb {R}^n)$
, we say that $H$
is (Gateaux) differentiable.
3. Continuity of the Hardy constant
Here we discuss some continuity results. By $co(\Omega )$
we denote the convex hull of $\Omega$
.
Theorem 3.1 Let $\Omega \subset \mathbb {R}^n$
be an open set with non-empty boundary, and let $\Sigma \subset \partial \Omega$
be an arbitrary subset of the boundary. Then there exist $\epsilon >0$
and $c>0$
such that for every $C^1$
diffeomorphism $\varphi$
with $\|D\varphi -I \|<\epsilon,$
where $co_\varphi (\Omega )=co(\Omega )\cup \varphi ^{-1}(co(\varphi (\Omega )))$
.
Proof. Let $u\in H^1_0(\Omega )$
be normalized by $\int _\Omega u^2/d_\Sigma ^2 {\rm d}x = 1$
. For $v=u\circ \varphi ^{-1}$
, consider the Rayleigh quotient
where the last equality follows from the change of variables $y=\varphi (x)$
. After some elementary calculations, it follows that
In order to get an estimate for the expression
we first note that $\| A^\top \| = \| A \|$
as operator norms. To get an upper bound for the operator norm of the inverse, we also make the assumption that $\varphi$
is a ‘small’ diffeomorphism in the sense that $D\varphi (x) = I+\epsilon (x)$
where $\| \epsilon (x) \|<1$
. In this case it is known that
Besides, for such $\epsilon$
there is a constant $\kappa =\kappa (n)$
such that
so eventually we have the estimate
for some constant $C>0$
provided that $\| D\varphi -I \|$
is small.
Next, for $x\in \Omega$
, we obtain an estimate of $d_{\varphi (\Sigma )}(\varphi (x))$
in terms of $d_\Sigma (x)$
. Since $d_\Sigma = d_{\bar {\Sigma }}$
, we may assume that $\Sigma$
is closed. Then there exists $\sigma (x) \in \Sigma$
such that $d_\Sigma (x)=|x-\sigma (x)|$
. Consider the straight line segment $\gamma :[0,\,1]\rightarrow \mathbb {R}^n$
,
joining these two points. Then clearly $d_\Sigma (x)=l(\gamma )$
(the arc length of $\gamma$
). Then, by definition, we have that
thus
It follows that
the last inequality being valid due to normalization, thus
for some constant $C$
provided that $\| D\varphi -I \|$
is small.
Using all these estimates we obtain
for some $c>0$
. Passing to the appropriate limit of minimizers, we get
Replacing $\Omega$
and $\Sigma$
by $\varphi (\Omega )$
and $\varphi (\Sigma )$
and $\varphi$
by $\varphi ^{-1}$
, it follows that
Since
it follows that there is $c>0$
such that the reverse inequality
also holds for small $\| D\varphi - I \|$
. The result follows.
For small $\| \varphi - id \|_{C^1}$
, we have that if $\Omega$
is relatively compact, so is $co_\varphi (\Omega )$
, so we immediately deduce the following.
Corollary 3.2 Let $\Omega \subset \mathbb {R}^n$
be open and bounded, and let $\Sigma \subset \partial \Omega$
. Then the map $\varphi \longmapsto H(\varphi (\Omega ),\,\varphi (\Sigma ))$
is continuous with respect to the weak $C^1$
topology.
A few remarks are in order. First, the result does not hold for the case $k=0$
(homeomorphisms), as it is essential to be able to control first derivatives. Next, note that estimate (3.1) holds independent of the boundedness of $\Omega$
or compactness of $\textrm {supp}(\varphi )$
, and is therefore substantially more general than the corollary.
Although of no use to the sequel, we now present a collateral result that is obtained without extra effort. Instead of the standard Euclidean distance $\textrm {dist}(x,\,y)=|x-y|$
, for $x,\,y\in \Omega$
one could use the alternative ‘interior’ distance
and consider the Hardy problem
where $\tilde {d}_\Sigma (x)=\widetilde {\textrm {dist}}(x,\,\Sigma )$
. For that case, we obtain the almost identical result
Theorem 3.3 Let $\Omega \subset \mathbb {R}^n$
be open set with non-empty boundary, and let $\Sigma \subset \partial \Omega$
be an arbitrary subset of the boundary. Then there exist $\epsilon >0$
and $c>0$
such that for every $C^1$
diffeomorphism $\varphi$
with $\|D\varphi -I \|<\epsilon,$
Proof. The proof is almost identical to that of estimate (3.1). The only difference is that instead of picking $\gamma$
to be the straight line segment joining $x$
and $\sigma (x)$
, one chooses a sequence of curves $\gamma _n$
such that $l(\gamma _n)\rightarrow \tilde {d}_\Sigma (x)$
.
Note that taking convex hulls is unnecessary here, since all distances are compared inside $\Omega$
.
4. Differentiability of the Hardy constant
Now we present our main results regarding differentiability. Our methodology is similar to the one developed in [Reference Barbatis and Lamberti5] (which concerns the case $\Sigma =\partial \Omega$
), with appropriate modifications.
Lemma 4.1 Suppose that $\Omega \subset \mathbb {R}^n$
is open with non-empty boundary and let $\Sigma \subset \partial \Omega$
be closed. Let $\varphi \in \textrm {Diff}^1_c(\mathbb {R}^n),$
$\xi \in \mathfrak {X}^1_c(\mathbb {R}^n)$
and let $t_0>0$
be such that
is a $C^1$
diffeomorphism for all $t\in [-t_0,\,t_0]$
. Then:
(i) There exists a constant $c=c(\Omega,\,\varphi,\,\xi,\,t_0)$
such that
(4.1)\begin{equation} |d_{\varphi_t(\Sigma)}^2(\varphi_t(x))-d_{\varphi(\Sigma)}^2(\varphi(x))| \leq c d_{\varphi(\Sigma)}^2(\varphi(x)) |t| \end{equation}for all $x\in \Omega$
and all $t\in [-t_0,\,t_0]$.(ii) If $d_{\varphi (\Sigma )}$
is differentiable at $\varphi (x)$ and $\sigma (x) \in \Sigma$ is the single point such that $d_{\varphi (\Sigma )}(\varphi (x))=|\varphi (x)-\varphi (\sigma (x))|,$ then
(4.2)\begin{equation} \frac{d}{{\rm d}t}\Big|_{t=0} d_{\varphi_t(\Sigma)}^2(\varphi_t(x)) = 2(\varphi(x)-\varphi(\sigma(x)))\cdot(\xi(x)-\xi(\sigma(x))). \end{equation}
Proof. (1) Let $x\in \Omega$
. Since $\Sigma$
is closed, there exists a $\sigma \in \Sigma$
such that $d_{\varphi (\Sigma )}(\varphi (x)) = |\varphi (x)-\varphi (\sigma )|$
. It follows that
Moreover, we have that
Likewise, let $\sigma _t \in \Sigma$
be such that $d_{\varphi _t(\Sigma )}(\varphi _t(x))=|\varphi _t(x)-\varphi _t(\sigma _t)|$
. Then
and as before we have
As $[-t_0,\,t_0]$
is compact, $\| D(\xi \circ \varphi _t^{-1}) \|_{L^\infty (co(\varphi _t(\Omega )))}$
attains a finite maximum value in it, and so follows the existence of a constant so that the conclusion holds.
(2) Assume that $d_{\varphi (\Sigma )}$
is differentiable at $\varphi (x)$. Thus there exists a unique $\sigma =\sigma (x) \in \Sigma$ such that $d_{\varphi (\Sigma )}(\varphi (x))=|\varphi (x)-\varphi (\sigma (x))|$. From (4.1), we know that
Now we claim that $\lim _{t\rightarrow 0} \sigma _t = \sigma$
($\sigma _t$
as defined in the previous step). To this end, it suffices to show that
Assume, by contradiction, that there exists $\sigma '\in \Sigma$
, $\sigma '\neq \sigma$
, such that, possibly passing to a subsequence,
Then
for some $\epsilon >0$
. In particular,
Moreover,
and by (4.3) we deduce that
a contradiction.
From the estimates of the previous step and the claim we deduce that
□
From this point on, we will assume that $\Omega$
is bounded and Lipschitz (in the sense of being locally the subgraph of a Lipschitz continuous function). By the results of [Reference Davies9], we know that the Hardy inequality holds in $\Omega$
for some positive constant for $\Sigma = \partial \Omega$
. Since $d_\Sigma \geq d_{\partial \Omega }$
, the same is true if we choose any $\Sigma \subset \partial \Omega$
.
Lemma 4.2 Suppose that $\Omega \subset \mathbb {R}^n$
is a bounded Lipschitz domain and let $\Sigma \subset \partial \Omega$
be closed. Let also $u\in H^1_0(\Omega )$
and $\rho \in L^\infty (\Omega )$
. Then the function $G: Diff^1_c(\mathbb {R}^n) \rightarrow \mathbb {R}$
$(k\geq 1)$
given by
is Gateaux differentiable and, for $\xi \in \mathfrak {X}^1_c(\mathbb {R}^n)$
Proof. Let $\varphi \in Diff^1_c(\mathbb {R}^n)$
and $\varphi _t=\varphi +t\xi$
as before. Then
By estimate (4.1), there is a constant $c>0$
such that
for $t$
sufficiently small. Since $\rho \in L^\infty (\Omega )$
and $\Omega$
is bounded, and since $u\in H^1_0(\Omega )$
and the Hardy inequality holds (the latter is true because $C^1$
diffeomorphisms preserve the Lipschitz propertyFootnote 1), it follows that the integrand is absolutely bounded by an $L^1$
function and the dominated convergence theorem applies.
Since $d_{\varphi (\Sigma )}(\varphi (x))$
is differentiable for almost all $x\in \Omega$
, the unique point $\sigma (x)\in \Sigma$
is defined for almost all $x\in \Omega$
and the result follows by (4.2).
We wish to prove that the Hardy constant $H(\varphi (\Omega ),\,\varphi (\Sigma ))$
is Gateaux differentiable with respect to $\varphi$
, which is equivalent to proving that the map $t\mapsto H(\varphi _t(\Omega ),\,\varphi _t(\Sigma ))$
is differentiable with respect to $t$
for any $\xi \in \mathfrak {X}^1_c(\mathbb {R}^n)$
, where
Doing so will be possible provided that there are actual minimizers to the constants $H(\varphi _t(\Omega ),\,\varphi _t(\Sigma ))$
, and that these actually behave ‘well’ as $t$
varies, i.e. they are stable.
Here we draw some important facts coming from other works that are vital in order to proceed. Note that since these results are stated for smooth (meaning $C^\infty$
) domains, we also adopt this assumption from now on, as stated in the Introduction.
Lemma 4.3 Suppose that $\Omega \subset \mathbb {R}^n$
$(n\geq 2)$
is a smooth bounded domain, and let $\Sigma \subset \partial \Omega$
be a closed submanifold of dimension $s\in \{0,\,1,\,\ldots,\,n-1 \}$
. Consider the Hardy problem
Then precisely one of the following alternatives is true:
(i) The problem has a minimizer and $H(\Omega,\,\Sigma )<(n-s)^2/4$
.(ii) The problem does not have a minimizer and $H(\Omega,\,\Sigma )=(n-s)^2/4$
.
Proof. This is corollary 1.3 in [Reference Fall and Mahmoudi11]. The case $s=0$
was treated separately in [Reference Fall and Musina12], and the case $s=n-1$
is well known (see [Reference Marcus, Mizel and Pinchover16]).
So in order to proceed we need from now on the additional assumption that $H(\varphi (\Omega ),\, \varphi (\Sigma )) < (n-s)^2/4$
in order to guarantee the existence of minimizers. This assumption is not terribly restrictive, since $\varphi \mapsto H(\varphi (\Omega ),\,\varphi (\Sigma ))$
is a continuous map and the inverse image of $(-\epsilon,\,(n-s)^2/4)$
with respect to that map is an open set of $Diff^1_c(\mathbb {R}^n)$
. This means That if $H(\Omega,\,\Sigma )$
is less than the critical value, there is always an open neighbourhood of the identity to conduct a local analysis. The case where $H(\Omega,\,\Sigma )$
attains the critical value remains an open problem, as in [Reference Barbatis and Lamberti5].
Next we provide some estimates for these minimizers.
Lemma 4.4 Let $\Omega$
and $\Sigma$
be as in the previous lemma, and suppose that $v\in H^1_0(\Omega )$
is a minimizer of (4.4). Then there is a constant $C=C(\Omega,\,\Sigma )>0$
such that
where
Proof. This was proved in [Reference Marcus and Nguyen17] for the eigenfunction corresponding to the first eigenvalue of the relevant Schrödinger operator (lemmas 2.1 and 2.2). The same steps can be repeated for $\lambda =0$
, which simplifies the proof even further.
Theorem 4.5 Suppose that $\Omega \subset \mathbb {R}^n$
$(n\geq 2)$
is a smooth bounded domain, and let $\Sigma \subset \partial \Omega$
be a closed submanifold of dimension $s$
. Let $v\in H^1_0(\Omega )$
be a minimizer of (4.4) (and so $H(\Omega,\,\Sigma )<(n-s)^2/4$
). Then the following estimates are satisfied:
where $C=C(\Omega,\,\Sigma )$
.
Proof. The first estimate is obvious from the previous lemma and the fact that $d_{\partial \Omega } \leq d_\Sigma$
.
For the second one we proceed as follows. Let $x\in \Omega$
and let $R=d_{\partial \Omega }(x)/3$
. Then for every $y\in B(x,\,R)$
we have that
At this point we invoke a gradient estimate such as
where $f=H(\Omega,\,\Sigma )v/d_\Sigma ^2$
, see for example [Reference Gilbarg and Trudinger14] (paragraph 3.4) for an analogue with cubes. Thus, after some elementary calculations, we get
where in each step constant factors are absorbed in $C$
.
Theorem 4.6 Let $\Omega \subset \mathbb {R}^n$
$(n\geq 2)$
be a smooth bounded domain, and let $\Sigma \subset \partial \Omega$
be a closed submanifold of dimension $s$
. Suppose that $H(\Omega,\,\Sigma )<(n-s)^2/4$
. Thus for any $\xi \in \mathfrak {X}^1_c(\mathbb {R}^n)$
, $\varphi _t=id+t\xi \in Diff^1_c(\mathbb {R}^n)$
and $H(\varphi _t(\Omega ),\,\varphi _t(\Sigma ))<(n-s)^2/4$
for $t$
small enough.
Let $v_t$
be a one-parameter family of positive minimizers for $H(\varphi _t(\Omega ),\,\varphi _t(\Sigma )),$
normalized by
and let $u_t = v_t \circ \varphi _t: \Omega \rightarrow \mathbb {R}$
. Then
Proof. By the normalization condition on the minimizers, it follows that $\| \nabla v_t \|_{L^2(\varphi _t(\Omega ))}=H(\varphi _t(\Omega ),\,\varphi _t(\Sigma ))$
, thus $\| v_t \|_{H^1_0(\varphi (\Omega ))}$
and $\| u_t \|_{H^1_0(\Omega )}$
are uniformly bounded. Hence, possibly passing to a subsequence, there is a $\tilde {u}_0$
such that
We will show that $\tilde {u}_0$
satisfies the same normalization condition, i.e.
This is actually a consequence of the dominated convergence theorem applied on
provided it is applicable. Indeed, from the previous estimates, we have that there are $C>0$
and $\alpha$
such that $2\alpha +n-s>0$
such that
uniformly in $t$
for $t$
small enough. It follows that
Choosing a tubular neighbourhood $\Sigma _\epsilon (t) = \{x\in \Omega : d_{\varphi _t(\Sigma )}(\varphi _t(x)) < \epsilon \}$
and passing to coordinates given by exponential mapping, for $r(x)=d_{\varphi _t(\Sigma )}(\varphi _t(x))$
we have that
where the integral of the right-hand side is convergent since $2\alpha +n-s-1>-1$
. Hence the integrand is uniformly bounded in $t$
by an integrable function, and the claim follows.
From vector inequality $|a|^2 \geq |b|^2+2b\cdot (a-b)$
, it follows that
By the dominated convergence theorem and the continuity of $H$
, it follows that
so $\tilde {u}_0$
must be a positive normalized minimizer, and by the uniqueness of such minimizers it follows that $\tilde {u}_0 = u_0$
.
Moreover, also by the dominated convergence theorem, we have that
so it follows that
Since weak convergence and convergence in norm imply strong convergence, the proof is complete.
Theorem 4.7 Let $\Omega \subset \mathbb {R}^n$
$(n\geq 2)$
be a smooth bounded domain, and let $\Sigma \subset \partial \Omega$
be a closed submanifold of dimension $s$
. Suppose that $H(\Omega,\,\Sigma )<(n-s)^2/4$
and let $v$
be a minimizer that achieves $H(\Omega,\,\Sigma )$
normalized by
Then the map $H: \varphi \mapsto H(\varphi (\Omega ),\,\varphi (\Sigma ))$
is Gateaux differentiable at $id_{\mathbb {R}^n}$
and
where $\sigma (x)$
is the (a.e. unique) point in $\Sigma$
such that $d_\Sigma (x)=|x-\sigma (x)|$
.
Proof. Let $\varphi _t = id + t\xi$
and $v_t$
a sequence of positive normalized minimizers as before. By the definition of the Hardy constant and change of variables, we have that
where $R_t[u]=N_t[u]/D_t[u]$
,
Since $v_t$
achieves $H(\varphi _t(\Omega ),\,\varphi _t(\Sigma ))$
, we have that $H(\varphi _t(\Omega ),\,\varphi _t(\Sigma )) = R_t[u_t]$
, where $u_t=v_t\circ \varphi _t$
as before.
It follows, by the definition of the Hardy constant, that
Now, $R_t[u]$
is a function of two arguments, a real number $t$
and a function $u$
. The partial derivative of this function with respect to $t$
is denoted by $R_t'[u]$
. The last inequality together with the mean value theorem on the first argument of $R$
imply that there are numbers $\xi (t)$
and $\eta (t)$
such that $|\xi (t)|,\,|\eta (t)| < |t|$
and
If we show that $R_{\xi (t)}'[u_t]t$
and $R_{\eta (t)}'[u_0]$
converge to the same number as $t\rightarrow 0$
, differentiability at $t=0$
is established. Some basic calculations reveal that
It follows that
and
By the dominated convergence theorem, it follows that
and by the dominated convergence theorem together with the previous stability result, we also have
and the claim is proved.
It remains to compute the derivative. We have
the last equality being valid due to normalization. The result immediately follows from the previous calculations, putting $t=0$
and taking into account that $u_0=v$
.
5. Differentiability with respect to boundary diffeomorphisms
Finally, we turn our attention to the matter of differentiability of the map
for $\varphi \in Diff^1(\partial \Omega )$
, where $Diff^1(\partial \Omega )$
is the group of diffeomorphisms of class $C^1$
of the smooth manifold $\partial \Omega$
Footnote 2. Note that in the case where $s=n-1$
, this problem is irrelevant since the boundary as a whole remains invariant under boundary diffeomorphisms, so in this sense it is new.
First we establish a continuity result. In particular, if $\varphi \in Diff^1(\partial \Omega )$
, the map $\varphi \mapsto H(\Omega,\, \varphi (\Sigma ))$
is shown to be continuous with respect to the $C^1$
topology.
Theorem 5.1 Let $\Omega \subset \mathbb {R}^n$
be a bounded open set with smooth non-empty boundary, and let $\Sigma \subset \partial \Omega$
be an arbitrary subset of the boundary. Then there exist $\epsilon >0$
and $c>0$
such that for any $\varphi \in Diff^1(\partial \Omega )$
satisfying $\| \varphi - id \|_{C^1(\partial \Omega )} <\epsilon,$
the estimate
holds.
Proof. This can actually be reduced to the first case. One simply needs to extend diffeomorphisms of the boundary to diffeomorphisms of the ambient space. This cannot be done for an arbitrary diffeomorphism, but for small diffeomorphisms it is achievable since $Diff^1(\partial \Omega )$
is locally contractible. Indeed, for $\| \varphi - id \|_{C^0}< inj(\partial \Omega )$
(the injectivity radius of $\partial \Omega$
is a positive number since $\partial \Omega$
is compact), define a homotopy $h:\partial \Omega \times [0,\,1] \rightarrow \partial \Omega$
,
where $\exp$
stands for the exponential mapFootnote 3 of $\partial \Omega$
as a Riemannian submanifold of $\mathbb {R}^n$
, while the assumption above ensures that $h(\cdot,\,t)$
remains a diffeomorphism for all $t$
.
We now pick a neighbourhood of $\partial \Omega$
that is diffeomorphic to $\partial \Omega \times (-\epsilon,\,\epsilon )$
, and a cut-off function $f:(-\epsilon,\,\epsilon )\rightarrow \mathbb {R}$
, $0\leq f\leq 1$
that is $1$
in a neighbourhood of $0$
. Then for $(x,\,y)\in \partial \Omega \times (-\epsilon,\,\epsilon )$
is a diffeomorphism of $\mathbb {R}^n$
with compact support that extends $\varphi$
(extend trivially outside the neighbourhood by the identity). We then apply (3.1) for $\Phi$
and the result follows from the fact that
which is obvious by the construction.
Similar to the Euclidean case, $Diff^k(\partial \Omega )$
has a differential structure that is locally homeomorphic to the Banach space $\mathfrak {X}^k(\partial \Omega )$
(note that here we need not take vector fields with compact support since $\partial \Omega$
is by assumption compact). The differential of a map $h:Diff^k(\partial \Omega ) \rightarrow \mathbb {R}$
at $\varphi \in Diff^k(\partial \Omega )$
along $\xi \in \mathfrak {X}^k(\partial \Omega )$
is given by
provided that the limit exists, where $\exp (t\xi ) \in Diff^k(\partial \Omega )$
is the map obtained by exponential mapping along $\xi$
, which is always a diffeomorphism for $t$
small enough due to compactness.
We finally show that the Hardy constant is Gateaux differentiable with respect to such boundary diffeomorphisms.
Theorem 5.2 Let $\Omega \subset \mathbb {R}^n$
be bounded and of smooth boundary. Then the map $h:Diff^1(\partial \Omega ) \rightarrow \mathbb {R},$
$\varphi \mapsto H(\Omega,\,\varphi (\Sigma ))$
is differentiable at all points where $H(\Omega,\,\varphi (\Sigma ))<(n-s)^2/4$
.
Proof. Without loss of generality, let $\varphi =id_{\partial \Omega }$
, and let $\xi \in \mathfrak {X}^1(\partial \Omega )$
. Then one can extend $\xi$
to a $\Xi \in \mathfrak {X}^1_c(\mathbb {R}^n)$
(e.g. using a standard argument involving partitions of unity). One can also assume that the support of $\Xi$
lies within a neighbourhood of the form $\partial \Omega \times (-\epsilon,\,\epsilon )$
, equipped with a metric such that $\partial \Omega$
is a totally geodesic submanifoldFootnote 4 . Then we have that
Since $\exp (t\xi )(\Omega )=\Omega$
and $\exp (t\Xi ) |_{\partial \Omega }=\exp (t\xi )$
, it follows that
where in the last equalities we regard $H$
as the function $\varphi \mapsto H(\varphi (\Omega ),\,\varphi (\Sigma ))$
as discussed in the previous section.
There is a particularly neat way to express this form of differentiability in the special case $s=0$
(a point boundary singularity).
Corollary 5.3 Let $\Omega \subset \mathbb {R}^n$
be bounded and of smooth boundary. Then the map $H:\partial \Omega \rightarrow \mathbb {R},$
$\sigma \mapsto H(\Omega,\,\{\sigma \})$
is differentiable at every $\sigma \in \partial \Omega$
where $H(\Omega,\,\{ \sigma \})< n^2/4$
.
Proof. The differential $D_\sigma H(\Omega,\,\{ \cdot \})(\xi )$
for $\xi \in T_\sigma \partial \Omega$
is
where $\gamma _\xi$
is a curve such that $\gamma _\xi (0) = \sigma$
and $\gamma _\xi '(0)=\xi$
. From now on assume that $\gamma _\xi$
is the unique geodesic with these properties. Choose a vector field $\Xi \in \mathfrak {X}(\partial \Omega )$
that extends $\xi$
. Then $\{ \gamma _\xi (t) \} = \exp (t\Xi )(\{ \sigma \})$
. It follows that
where $h(\varphi )=H(\Omega,\,\varphi (\{ \sigma \}))$
as before.
Acknowledgements
Special thanks are owed to my PhD supervisor, Professor G. Barbatis, for the time he spent reviewing the article and offering useful suggestions. This research was supported by the Hellenic Foundation for Research and Innovation (HFRI) under the HFRI PhD Fellowship grant (Fellowship Number 1250).
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