# Functional inequalities: nonlinear flows and entropy methods as a tool for
obtaining sharp and constructive results

Jean Dolbeault  CEREMADE (CNRS UMR n∘ 7534), PSL university, Université Paris-
Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
<EMAIL_ADDRESS>
###### Abstract.

Interpolation inequalities play an essential role in Analysis with fundamental
consequences in Mathematical Physics, Nonlinear Partial Differential Equations
(PDEs), Markov Processes, _etc._ , and have a wide range of applications in
various other areas of Science. Research interests have evolved over the
years: while mathematicians were originally focussed on abstract properties
(for instance appropriate notions of functional spaces for the existence of
weak solutions in PDEs), more qualitative questions (for instance, bifurcation
diagrams, multiplicity of the solutions in PDEs and their qualitative
behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs
is a typical example: in some cases, the optimal constant in the inequality
can be interpreted as an optimal rate of decay of an entropy for an associated
evolution equation. Much more has been learned by adopting this point of view.

This paper aims at illustrating some of these recent aspect of entropy-entropy
production inequalities, with applications to stability in Gagliardo-
Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-
Nirenberg inequalities. Entropy methods provide a framework which relates
nonlinear regimes with their linearized counterparts. This framework allows to
prove optimality results, symmetry results and stability estimates. Some
emphasis will be put on the hidden structure which explain such properties.
Related open problems will be listed.

###### Key words and phrases:

Interpolation, Gagliardo-Nirenberg inequality, Caffarelli-Kohn-Nirenberg
inequality, stability, entropy methods, entropy-entropy production inequality,
carré du champ, fast diffusion equation, Harnack Principle, asymptotic
behaviour, Hardy-Poincaré inequalities, spectral gap, intermediate
asymptotics, self-similar Barenblatt solutions, rates of convergence,
symmetry, symmetry breaking, bifurcation

###### 1991 Mathematics Subject Classification:

Primary: 46E35; 35K55; 35B06. Secondary: 26D10; 49J40; 35B40; 49K20; 49K30;
35J60; 35J20; 53C21.

The two main results of Sections 2 and 3 are based on recent papers. Stability
results on Gagliardo-Nirenberg-Sobolev inequalities have been obtained in
collaboration with M. Bonforte, B. Nazaret, and N. Simonov and are collected
in [20]. Symmetry results in Caffarelli-Kohn-Nirenberg inequalities rely on a
series of papers with M.J. Esteban, M. Loss and M. Muratori, see [52, 18, 55].
The guideline of this paper is to clarify the interplay of nonlinear
interpolation inequalities with their linearized counterparts using entropy
methods and fast diffusion flows.

##  1\. A brief historical perspective

The goal of this section is to sketch important steps in the development of
the theory of some important functional inequalities, in the perspective of
entropy methods. It is neither a complete review nor a detailed historical
account, but more a personal selection of some results which are highlighted
as representative of the evolution of the ideas. Several results have been
discovered at least twice, in completely independent papers. Whenever
possible, this has been clarified, but such issues are always delicate and the
overall information is probably still incomplete.

###  1.1. Sobolev and some other interpolation inequalities

Unless it is specified, we consider functions on ${\mathbb{R}}^{d}$. For any
$q\in[1,\infty)$,
$\|{f}\|_{q}:=\left(\int_{{\mathbb{R}}^{d}}{|f|^{q}}\,dx\right)^{1/q}$, whilst
$\|{f}\|_{\infty}$ denotes the $\mathrm{L}^{\infty}({\mathbb{R}}^{d})$ norm.

####  1.1.1. Sobolev inequality

On ${\mathbb{R}}^{d}$ with $d\geq 3$, the classical _Sobolev inequality_
asserts that

$\|{\nabla f}\|_{2}^{2}\geq\mathsf{S}_{d}\,\|{f}\|_{2^{*}}^{2}$ (1.1)

for any smooth and compactly supported function, where $2^{*}=2\,d/(d-2)$ is
the critical exponent and the optimal constant is

$\mathsf{S}_{d}=\pi\,d\,(d-2)\,\Big{(}\tfrac{\Gamma\left(d/2\right)}{\Gamma\left(d\right)}\Big{)}^{2/d}=\frac{1}{4}\,d\,(d-2)\,\frac{2^{\frac{2}{d}}\,\pi^{1+\frac{1}{d}}}{\Gamma\big{(}(d+1)/2\big{)}^{2/d}}=\frac{1}{4}\,d\,(d-2)\,\big{|}\mathbb{S}^{d}\big{|}^{\frac{2}{d}}\,.$

Inequality (1.1) is easily extended to the Beppo-Levi space using standard
completion results with respect to the norm $f\mapsto\left(\|{\nabla
f}\|_{2}^{2}+\|{f}\|_{2^{*}}^{2}\right)^{1/2}$, see for instance [44].
Inequality (1.1) appears in [90] and is extremely well understood: see [6, 92,
29]. The question of the optimality of the constant $\mathsf{S}_{d}$ has
anyway a much longer history.

Among radial functions and at least in dimension $d=3$ for the critical
exponent $2^{*}=6$, the solutions of the Euler-Lagrange equation enter in the
study of the polytropic and isothermal gas spheres, which are models of
gaseous stars in astrophysics and can be traced back to the work of H. Lane in
[74]. A detailed account can be found in [36, Chapter IV]. The explicit
computation of the optimal radial functions and the corresponding value of
$\mathsf{S}_{d}$ appears in [16]. Further considerations on the linear
stability of the radial optimal functions can be found in [87].

The classical papers for the optimal Sobolev inequality are those of T. Aubin
and G. Talenti [6, 91], with an earlier contribution by E. Rodemich in [86]
which is quoted for instance in [66, p. 158]. These papers rely on
_spherically symmetric rearrangements_ and particularly on the Pólya–Szegő
principle: we refer to [72, 73, 93] and references therein for details. An
alternative proof based on Riesz’ lemma and duality can be found in [76, 78].
Many other proofs have been established using various techniques like, _e.g._
, mass transport in [40]. Some results based on nonlinear flows and _entropy
methods_ , which completely avoid spherically symmetric rearrangements, will
be exposed in Section 2.

Optimal functions in (1.1) are known as the _Aubin-Talenti functions_. The set
$\mathfrak{M}$ of these functions is a $(d+2)$-dimensional manifold generated
by dilations, translations and multiplications by a constant of

$\mathsf{g}(x):=\left(1+|x|^{2}\right)^{-(d-2)/2}\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

Stability is a delicate issue. As an answer to some of the questions raised in
[30], G. Bianchi and H. Egnell proved in [13] the existence of $\mathcal{C}>0$
such that

$\|{\nabla
f}\|_{2}^{2}-\mathsf{S}_{d}\,\|{f}\|_{2^{*}}^{2}\geq\mathcal{C}\,\inf_{g\in\mathfrak{M}}\|{\nabla
f-\nabla g}\|_{2}^{2}\,,$ (1.2)

The existence of $\mathcal{C}$ is obtained by concentration-compactness
methods and by contradiction, hence comes with no estimate. Various
refinements starting with [38] and the more recent results of [61, 82, 62]
have been obtained, although without explicit and constructive estimates on
$\mathcal{C}$. By duality methods inspired by [77] and flows, quantitative and
constructive results were obtained in weak norms in [45, 56]. In [20], a
result of stability has been obtained, with a measure of the distance to
$\mathfrak{M}$ by a relative Fisher information, a strong norm: this result
will be presented in Section 2.

####  1.1.2. Gagliardo-Nirenberg inequalities

In two papers, [64, 83], E. Gagliardo and L. Nirenberg established a number of
interpolation inequalities which are now famous. Some historical details can
be found in [84]. Here we are interested in the sub-family

$\|{\nabla
f}\|_{2}^{\theta}\,\|{f}\|_{p+1}^{1-\theta}\geq\mathcal{C}_{\mathrm{GNS}}(p)\,\|{f}\|_{2p}$
(1.3)

where the exponent $\theta=\frac{d\,(p-1)}{(d+2-p\,(d-2))\,p}$ is fixed by
scaling considerations and functions $f$ are taken in the space obtained as
the completion of the space of infinitely differentiable functions on
${\mathbb{R}}^{d}$ with compact support, with respect to the norm
$f\mapsto(1-\theta)\,\|{f}\|_{p+1}+\theta\,\|{\nabla f}\|_{2}$. We shall say
that the exponent $p$ is an _admissible exponent_ if

$p\in(1,+\infty)\quad\mbox{if}\quad d=1\;\mbox{or}\;2\,,\quad
p\in(1,p^{\star}]\quad\mbox{if}\quad d\geq 3\quad\mbox{with}\quad
p^{\star}:=\tfrac{d}{d-2}\,.$

In the limit case where $p=p^{\star}$, $d\geq 3$, for which $\theta=1$, we
notice that $2\,p^{\star}=2^{*}$ and recover (1.1) as a limit case of (1.3).
Other interesting limits can be obtained: as $p\to 1$, Inequality (1.3)
degenerates according to [41] into the Euclidean logarithmic Sobolev
inequality in scale invariant form, while in the limit as $p\to+\infty$ when
$d=2$ (see for instance [42]), we recover the _Euclidean Onofri inequality_.
See [20, Chapter 1] for more details. Inequality (1.3) was proved in [41].
Actually, in [67], an earlier result was stated with a rather complete scheme
of a proof, up to the delicate question of the uniqueness. None of the authors
of [41] were aware of [67] but they had by the time the paper was written fine
uniqueness results available from the mathematical literature. Entropy methods
are now providing alternative tools which have been exploited in [20, Chapter
1]. In any case, the key issue is that optimal functions in (1.3) can be
identified. They are given up to dilations, translations and multiplications
by a constant by

$\mathsf{g}(x):=\left(1+|x|^{2}\right)^{-\frac{1}{p-1}}\quad\forall\,x\in{\mathbb{R}}^{d}\,.$
(1.4)

Hence they generalize the _Aubin-Talenti functions_. For this reason, we shall
still use this denomination and denote again by $\mathfrak{M}$ the
corresponding manifold. Actually there are deeper reasons for that than the
algebraic expression of $\mathsf{g}$, that were clarified in [9, Section 6.10]
(also see [20, Section 1.3.1]). Stability results for (1.3) is one of the main
achievement of entropy methods based on nonlinear diffusions and will detailed
in Section 2 (see Theorems 2.5 and 2.6).

####  1.1.3. Caffarelli-Kohn-Nirenberg inequalities

The critical inequality

$\left(\int_{{\mathbb{R}}^{d}}{\frac{|f|^{p}}{|x|^{b\,p}}}\,dx\right)^{2/p}\leq\,\mathsf{C}_{a,b}\int_{{\mathbb{R}}^{d}}{\frac{|\nabla
f|^{2}}{|x|^{2\,a}}}\,dx$ (1.5)

is valid for any function
$f\in\left\\{\,f\in\mathrm{L}^{p}\big{(}{\mathbb{R}}^{d},|x|^{-b}\,dx\big{)}\,:\,|x|^{-a}\,|\nabla
f|\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d},dx\big{)}\right\\}$ under the
conditions that $a\leq b\leq a+1$ if $d\geq 3$, $a<b\leq a+1$ if $d=2$,
$a+1/2<b\leq a+1$ if $d=1$, and $a<a_{c}:=(d-2)/2$ where the exponent
$p=2\,d/(d-2+2\,(b-a))$ is again fixed by scaling considerations. Here
$\mathsf{C}_{a,b}$ denotes the optimal constant. Inequality (1.5) was
apparently introduced first by V.P. Il’in in [69] but is more known in the
literature as the _critical Caffarelli-Kohn-Nirenberg inequalities_ ,
according to [33]. An important step in the understanding of this inequality
came with [34] where the question of the _symmetry breaking_ problem was
raised: in (1.5), is optimality achieved among radially symmetric functions ?
Notice that partial symmetry results were known before, see, _e.g._ , [37,
68]: the point in [34] (also see [89]) was to provide a mechanism for proving
that symmetry breaking also occurs. If symmetry occurs, it can be proved that
the equality case is achieved by

$\mathsf{g}(x):=\left(1+|x|^{(p-2)(a_{c}-a)}\right)^{-\frac{2}{p-2}}\quad\forall\,x\in{\mathbb{R}}^{d}$
(1.6)

and the optimal constant is then given by

$\mathsf{C}_{a,b}=\tfrac{2}{p}\,((a_{c}-a)^{-\frac{p+2}{p}}\left(\frac{(p-2)\,\Gamma\big{(}\frac{3\,p-2}{2\,(p-2)}\big{)}}{2\,\sqrt{\pi}\,\big{|}\mathbb{S}^{d-1}\big{|}\;\Gamma\big{(}\frac{p}{p-2}\big{)}}\right)^{\frac{p-2}{p}}\,.$

Assuming that this was the case, V. Felli and M. Schneider systematized the
strategy of [34] in [60]: by linearizing the functional made of the difference
of the two sides in (1.5), they found a negative eigenvalue if $b<b_{\rm
FS}(a)$ with

$b_{\rm FS}(a):=\frac{d\,(a_{c}-a)}{2\,\sqrt{(a_{c}-a)^{2}+d-1}}+a-a_{c}\,,$

thus proving the linear instability of $\mathsf{g}$, a contradiction. As a
consequence, _symmetry breaking_ occurs if $b<b_{\rm FS}(a)$. Reciprocally,
one of the major achievements of entropy methods is the following symmetry
result.

######  Theorem 1.1 (Symmetry _vs._ symmetry breaking, critical case, [52]).

Assume that $d\geq 2$, $a<a_{c}$, and $b_{\rm FS}(a)\leq b\leq a+1$ if $a<0$.
Then equality in (1.5) is achieved if and only if, up to a scaling and a
multiplication by a constant, $f=\mathsf{g}$.

The result in [52] is actually stronger: it is proved that there is no other
critical point of the Euler-Lagrange equation, which is a _rigidity result_.
The method is not limited to critical inequalities and there is also a family
of _subcritical Caffarelli-Kohn-Nirenberg inequalities_ , exactly as in the
case without weights. Similar results have been obtained in [18, 19, 55] for
the inequality

$\left(\int_{{\mathbb{R}}^{d}}{\frac{|f|^{2\,p}}{|x|^{\gamma}}}\,dx\right)^{1/p}\leq\,\mathsf{C}_{\beta,\gamma,p}\left(\int_{{\mathbb{R}}^{d}}{\frac{|\nabla
f|^{2}}{|x|^{\beta}}}\,dx\right)^{\theta}\left(\int_{{\mathbb{R}}^{d}}{\frac{|f|^{p+1}}{|x|^{\gamma}}}\,dx\right)^{1-\theta}$
(1.7)

with optimal constant $\mathsf{C}_{\beta,\gamma,p}$, parameters $\beta$,
$\gamma$ and $p$ such that

$\gamma-2<\beta<\frac{d-2}{d}\,\gamma\,,\quad\gamma\in(-\infty,d)\,,\quad
p\in\left(1,p_{\star}\right]\quad\mbox{with}\quad
p_{\star}:=\frac{d-\gamma}{d-\beta-2}\,,$ (1.8)

and an exponent
$\theta=\frac{(d-\gamma)\,(p-1)}{p\,(d+\beta+2-2\,\gamma-p\,(d-\beta-2))}$
which is determiend by scaling considerations. The admissible set of
parameters $(\beta,\gamma)$ is restricted by the condition $p\leq p_{\star}$,
which means

$\beta\geq d-2+\frac{\gamma-d}{p}\,.$

If symmetry holds, then the equality case in (1.7) is achieved by

$\mathsf{g}(x):=\left(1+|x|^{2+\beta-\gamma}\right)^{-\frac{1}{p-1}}\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

The following results are taken from [18, Theorem 2] and in [55, Theorem 1.1].

######  Theorem 1.2 (Symmetry _vs._ symmetry breaking, critical case, [18,
55]).

Assume that $d\geq 2$ and $\beta$, $\gamma$ and $p$ satisfy (1.8). Symmetry
holds in (1.7) if and only if

$\gamma<d\,,\quad\gamma-2<\beta<\frac{d-2}{d}\,\gamma\quad\mbox{and}\quad\beta\leq\beta_{\rm
FS}(\gamma):=d-2-\sqrt{(\gamma-d)^{2}-4\,(d-1)}\,.$

###  1.2. Branches of solutions

In 1983, H. Brézis and L. Nirenberg published a paper, [31], devoted to the
positive solutions of the equation

$-\Delta f+\lambda\,f=f^{p-1}$ (1.9)

in bounded domains, with homogeneous Dirichlet boundary conditions. They were
interested in the case of the critical exponent $p=2^{*}$, $d\geq 3$, and
related issues of compactness in connection with the Yamabe problem. This
paper had a considerable impact. With [63], it was a starting point for an
amazing effort to construct various solutions by variational methods or using
Lyapunov-Schmidt reduction techniques, not only on the Euclidean space but
also on Riemannian manifolds (where $\Delta$ has to be understood as the
Laplace-Beltrami operator). One of the typical issues is to understand how
branches of solutions depend on the parameter $\lambda$ and at which values
bifurcations may eventually occur, as part of the larger question of the
multiplicity of the solutions and their classification. Non-existence
(typically based on Pohožaev’s identity) and uniqueness results are another
related issue, with a series of prominent papers, [39, 65, 14], that can now
be reinterpreted using entropy methods. In [14] for instance, it is proved
that (1.9) written on the sphere $\mathbb{S}^{d}$ has a _unique_ positive
solution, $f\equiv\lambda^{-1/(p-2)}$, if and only if $\lambda\leq d/(p-2)$.
As a consequence, according to [14, Corollary 6.1] (also see [11]), we obtain
the inequality

$\|{\nabla
f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\frac{d}{p-2}\left(\|{f}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\quad\forall\,f\in\mathrm{H}^{1}(\mathbb{S}^{d},d\mu)$
(1.10)

where $d\mu=\big{|}\mathbb{S}^{d}\big{|}^{-1}\,dv_{g}$ denotes the uniform
probability measure on the sphere and $p\in(2,2^{*})$, with the convention
that $2^{*}=+\infty$ if $d=1$ or $2$. Inequality (1.10) has been studied in
[14] by _rigidity_ methods, in [11] by techniques of harmonic analysis, and
using the _carré du champ_ method in [12, 10, 43], for any $p>2$, with
improvements and extensions in [49, 53]. We refer to [54] for a review and
[48] for most recent results.

If $d\geq 3$, the case $p=2^{*}$ is also covered: Inequality (1.10) is the
Sobolev inequality on the sphere and it is equivalent to (1.1), up to a
stereographic projection. Taking into account the normalization of $d\mu$, it
is easy to recover the expression of $\mathsf{S}_{d}$. Another limit case is
the _logarithmic Sobolev inequality_ on $\mathbb{S}^{d}$,

$\|{\nabla
f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\frac{d}{2}\,\int_{\mathbb{S}^{d}}{|f|^{2}\,\log\left(\frac{|f|^{2}}{\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}\right)}\,d\mu\quad\forall\,f\in\mathrm{H}^{1}(\mathbb{S}^{d},d\mu)\setminus\\{0\\}\,,$
(1.11)

which corresponds to the limit as $p\to 2_{+}$. Remarkably, (1.10) is also
valid for any $p\in[1,2)$ where the case $p=1$ corresponds simply to the
Poincaré inequality on the sphere. In this range, and actually for any
$p\in[1,2)\cup(2,+\infty)$ if $d=1$ or $p\in[1,2)\cup(2,2^{\\#}]$ if $d\geq
2$, where

$2^{\\#}:=\frac{2\,d^{2}+1}{(d-1)^{2}}$

is the _Bakry-Emery exponent_ , Inequalities (1.10) and (1.11) were proved
earlier by the _carré du champ_ method of D. Bakry and M. Emery in [7].

From the point of view of elliptic PDEs, the _carré du champ_ method amounts
to test (1.9) by $\Delta f+(p-1)\,|\nabla f|^{2}/f$. After using the Bochner-
Lichnerowicz-Weitzenböck formula

$\tfrac{1}{2}\,\Delta\,(|\nabla
f|^{2})=|\mathrm{Hess}f|^{2}+\nabla\cdot(\Delta f)\cdot\nabla
f+\mathrm{Ric}(\nabla f,\nabla f)\,,$ (1.12)

the fact that $\mathrm{Ric}(\nabla f,\nabla f)=(d-1)\,|\nabla f|^{2}$ on
$\mathbb{S}^{d}$, and a few integrations by parts, we obtain the identity

$\int_{\mathbb{S}^{d}}{\left(\tfrac{d}{d-1}\,|Q_{f}|^{2}+(d-\lambda)\,|\nabla
f|^{2}+\gamma(p)\,\frac{|\nabla f|^{4}}{f^{2}}\right)}\,d\mu=0$

with

$Q_{f}:=\mathrm{Hess}f-\frac{1}{d}\,\Delta
f\,\mathrm{Id}-\frac{d-1}{d+2}\,(p-1)\left(\frac{\nabla f\otimes\nabla
f}{f}-\frac{1}{d}\,\frac{|\nabla f|^{2}}{f}\,\mathrm{Id}\right)$ (1.13)

and

$\gamma(p):=\left(\frac{d-1}{d+2}\right)^{2}(p-1)\,(2^{\\#}-p)\,.$ (1.14)

Here we use the notation
$|\mathsf{A}|^{2}=\sum_{i,j=1}^{d}\mathsf{A}_{ij}^{2}$ if
$\mathsf{A}=(\mathsf{A}_{ij})_{i,j=1}^{d}$ is a $d\times d$ matrix. It is
therefore clear that the unique positive solution of (1.9) is a constant if
$\lambda\leq d$. This result has an interesting consequence in terms of
branches of solutions. On $\mathrm{H}^{1}(\mathbb{S}^{d},d\mu)$, the
functional

$f\mapsto\|{\nabla
f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{\lambda}{p-2}\left(\|{f}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)$

always admits $f\equiv 1$ as a critical point, there is no other positive
solution of (1.9) if $\lambda\leq d$ and the functional is nonnegative. For
any $\lambda>d$, a simple linearization around the constant solution shows
that there is a minimizer which makes the functional negative and $\lambda=d$,
which corresponds to the first positive eigenvalue of the Laplace-Beltrami
operator, is a bifurcation point. Let us define

$\mu(\lambda):=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d})\setminus\\{0\\}}\frac{(p-2)\,\|{\nabla
f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\lambda\,\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}{\|{f}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}}\,.$

The uniqueness result in the range $\lambda\in[0,d]$ and the bifurcation
result at $\lambda=d$ have the following consequences. If $\lambda\leq d$,
then $\mu(\lambda)=\lambda$, while $\mu(\lambda)<\lambda$ if $\lambda>d$. See
Fig. 1.

$\lambda$$\mu$$\mu=\mu(\lambda)$$\mu=\lambda$ Figure 1. Numerical computation
of the bifurcation diagram for $d=3$ and $p=3$. The branch emerging at
$\lambda=d$ from the straight line $\mu=\lambda$ of the constant solutions is
expected to realize the infimum $\lambda\mapsto\mu(\lambda)$.

The carré du champ method is limited to the range $p\leq 2^{\\#}$, but the
cases $p\in(2^{\\#},+\infty)$ if $d=1$ or $2$, or $p\in(2^{\\#},2^{*}]$ if
$d\geq 3$ can be dealt with using another test function, as we shall see
below: see [14, 50, 53] and references therein for more details. The method
admits various extensions, for instance to smooth compact connected Riemannian
manifolds, or improvements which can be used in order to obtain stability
estimates. For instance, on ${\mathbb{S}}^{d}$, the inequality

$\|{\nabla
f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\frac{d}{2-p-\gamma}\left(\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\|{f}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2-\frac{2\,\gamma}{2-p}}\,\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{\frac{2\,\gamma}{2-p}}\right)\quad\forall\,f\in\mathrm{H}^{1}(\mathbb{S}^{d})$
(1.15)

is established in [48] for $d\geq 1$, $p\in[1,2)\cup(2,2^{\\#})$ with the
convention $2^{\\#}=+\infty$ if $d=1$ and $\gamma=\gamma(p)$ given by (1.14)
if $d\geq 3$ and $\gamma(p)=(p-1)/3$ if $d=1$, under the condition that
$\gamma(p)\neq 2-p$. If $\gamma(p)=2-p$, there is a logarithmic version of the
inequality. Inequality (1.15) can be used to obtain estimates on the branches
of solutions as in [48, Theorem 2].

A major difficulty in the elliptic point of view of [39, 65] is the
construction of the appropriate test functions. Reinterpreted in terms of
adapted entropy methods and gradient flow estimates, building the estimates
becomes a much simpler task. This is what we will explain next.

###  1.3. Entropies and _carré du champ_ methods

_Carré du champ_ methods were introduced in [8] by D. Bakry and M. Emery and
have been successfully applied to a wide range of questions: a general
overview can be found in [9]. Here we give two simple examples, first in the
classical setting of linear diffusions with a confinement potential on
${\mathbb{R}}^{d}$, and then we revisit the inequalities (1.10) of Section 1.2
dealing with interpolation on $\mathbb{S}^{d}$.

####  1.3.1. Inequalities, rates of convergence and entropy methods for linear
diffusions

On ${\mathbb{R}}^{+}\times{\mathbb{R}}^{d}$, let us consider the _heat
equation_

$\frac{\partial u}{\partial t}=\Delta u\,,\quad u(t=0,x)=u_{0}(x)$ (1.16)

and assume that $u_{0}\in\mathrm{L}^{1}\cap\mathrm{L}^{2}({\mathbb{R}}^{d})$
is nonnegative. Among various classical results, it is well known that as
$t\to+\infty$, the solution behaves like the Green function
$G(t,x)=(4\,\pi\,t)^{-d/2}\,\exp(-|x|^{2}/(4\,t))$. As a consequence, we know
that $\|{u(t,\cdot)}\|_{2}^{2}=O(t^{-d/2})$ as $t\to+\infty$, which is easily
recovered using Nash’s inequality:

$\|{u}\|_{2}^{2+\frac{4}{d}}\leq\mathcal{C}_{\rm
Nash}\,\|{u}\|_{1}^{\frac{4}{d}}\,\|{\nabla u}\|_{2}^{2}$ (1.17)

For a solution of (1.16), we know that
$\|{u(t,\cdot)}\|_{1}=\|{u_{0}}\|_{1}=:M$ for any $t\geq 0$ and can indeed
write that

$\frac{d}{dt}\|{u(t,\cdot)}\|_{2}^{2}=-\,2\,\|{\nabla
u(t,\cdot)}\|_{2}^{2}\leq-\,\frac{2}{\mathcal{C}_{\rm
Nash}}\,\frac{\|{u(t,\cdot)}\|_{2}^{2+4/d}}{M^{4/d}}$

and the decay estimate easily follows after integrating with respect to $t$. A
slightly more delicate issue is to obtain an _intermediate asymptotics_
estimate, for instance an estimate of $u(t,\cdot)-M\,G(t,\cdot)$ in
$\mathrm{L}^{1}({\mathbb{R}}^{d})$. This can be done with entropy methods as
follows. Using the change of variables

$u(t,x)=R(t)^{-d}\,v\left(\log
R(t),\tfrac{x}{R(t)}\right)\quad\mbox{with}\quad
R(t)=\sqrt{1+2\,t}\quad\forall\,(t,x)\in{\mathbb{R}}^{+}\times{\mathbb{R}}^{d}\,,$

solving (1.16) is equivalent to solve the _Fokker-Planck equation_

$\frac{\partial v}{\partial t}=\Delta v+\nabla\cdot(x\,v)\,,\quad
v(t=0,x)=u_{0}(x)\,.$ (1.18)

A remarkable fact is that (1.18) admits a stationary solution $M\,\mu$ with
$\mu(x):=G(1/2,x)$. Up to an additional change of unknown function, the
function $w(t,x)=v(t,x)/\mu(x)$ solves the _Ornstein-Uhlenbeck equation_

$\frac{\partial w}{\partial t}=\Delta w+x\cdot\nabla w\,,\quad
w(t=0,x)=\frac{u_{0}(x)}{\mu(x)}\,.$ (1.19)

With the Gaussian measure $d\mu=\mu(x)\,dx$, an elementary computations shows
that

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{\left|w-M\right|^{2}}\,d\mu=-\,2\int_{{\mathbb{R}}^{d}}{\left|\nabla
w\right|^{2}}\,d\mu\leq-\,2\int_{{\mathbb{R}}^{d}}{\left|w-M\right|^{2}}\,d\mu$

where the last inequality is a simple consequence of the _Gaussian Poincaré
inequality_

$\int_{{\mathbb{R}}^{d}}{\left|f-\bar{f}\right|^{2}}\,d\mu\leq\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\quad\mbox{where}\quad\bar{f}:=\int_{{\mathbb{R}}^{d}}{f}\,d\mu\,,\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d})\,.$

It seems that this inequality was explicitly written in that form in [81], in
the very same paper in which J. Nash proved (1.17). With this observation and
a Grönwall estimate, we obtain the exponential decay of
$\int_{{\mathbb{R}}^{d}}{\left|w-M\right|^{2}}\,d\mu\geq\left(\int_{{\mathbb{R}}^{d}}{\left|w-M\right|}\,d\mu\right)^{2}$,
which controls $\|{u(t,\cdot)-M\,G(t,\cdot)}\|_{1}$. Alternatively, one can
directly consider the _entropy_ and compute

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{w\,\log(w/M)}\,d\mu=-\int_{{\mathbb{R}}^{d}}{w\,|\nabla\log
w|^{2}}\,d\mu\leq-\,2\int_{{\mathbb{R}}^{d}}{w\,\log(w/M)}\,d\mu$

where the last inequality is a simple consequence of the _Gaussian logarithmic
Sobolev inequality_

$\int_{{\mathbb{R}}^{d}}{f^{2}\,\log(f^{2}/M)}\,d\mu\leq
2\int_{{\mathbb{R}}^{d}}{|\nabla f|^{2}}\,d\mu\quad\mbox{where}\quad
M=\int_{{\mathbb{R}}^{d}}{f^{2}}\,d\mu\,,\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d})\,.$

Using again a Grönwall estimate and the Pinsker-Csiszár-Kullback inequality

$\int_{{\mathbb{R}}^{d}}{w\,\log(w/M)}\,d\mu\geq\frac{1}{4\,M}\left(\int_{{\mathbb{R}}^{d}}{|w-M|}\,d\mu\right)^{2}\,,$

we can conclude as before. It turns out that the heat equation can be seen as
the gradient flow of the entropy $\int_{{\mathbb{R}}^{d}}{u\,\log u}\,dx$ with
respect to Wasserstein’s distance as shown in [70] and the entropy formulation
turns out to be better adapted when linear diffusion are replaced by nonlinear
diffusions. For more details on Nash’s inequality, consequences for the heat
equation, the comparison of the decay estimates and more detailed references,
we refer to [20, 28].

A natural extension of the above heat related estimates is to consider linear
diffusions with confinement due to a given potential $\phi(x)$, which amounts
to study the _Ornstein-Uhlenbeck equation_ with potential $\phi$, that is,

$\frac{\partial w}{\partial t}=\Delta w-\nabla\phi\cdot\nabla w\,.$ (1.20)

This equation is reduced to (1.19) if $\phi(x)=|x|^{2}/2$. The invariant
measure is now $d\mu=Z^{-1}\,e^{-\phi}\,dx$ with
$Z=\int_{{\mathbb{R}}^{d}}{e^{-\phi}}\,dx$ and the _Ornstein-Uhlenbeck
operator_ $\mathcal{L}:=\Delta-\nabla\phi\cdot\nabla$ is self-adjoint on
$\mathrm{L}^{2}({\mathbb{R}}^{d},d\mu)$ in the sense that

$\int_{{\mathbb{R}}^{d}}{w_{1}\,\mathcal{L}w_{2}}\,d\mu=-\int_{{\mathbb{R}}^{d}}{\nabla
w_{1}\cdot\nabla w_{2}}\,d\mu$

As seen above, obtaining convergence rates heavily relies on functional
inequalities. One of the key features of entropy methods is that the flow of
(1.20) can be used to establish these inequalities by the _carré du champ_
method, under the assumption that

$\mathrm{Hess}\,\phi\geq\kappa\,\mathrm{Id}\quad\mbox{a.e.}$ (1.21)

for some $\kappa>0$. Under this assumption, we claim that

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\geq\frac{\kappa}{p-2}\left(\left(\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu\right)^{2/p}-\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)$
(1.22)

for any $p\in[1,2)$, and also, by taking the limit as $p\to 2_{-}$ and with
the notation $M=\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu$,

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\geq\frac{\kappa}{2}\int_{{\mathbb{R}}^{d}}{|f|^{2}\,\log\left(|f|^{2}/M\right)}\,d\mu\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)\,.$

If $w$ solves (1.20) with initial datum $w_{0}=|f|^{p}$, the strategy of the
_carré du champ_ method is to notice that
$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{w}\,d\mu=0$ and that $w^{2}$ converges to
the constant $M=\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu$ while

$\frac{d}{dt}\left(\int_{{\mathbb{R}}^{d}}{\left|\nabla
w^{1/p}\right|^{2}}\,d\mu+\frac{\kappa}{p-2}\int_{{\mathbb{R}}^{d}}{w^{2/p}}\,d\mu\right)\leq
0\,,$

so that

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu-\frac{\kappa}{p-2}\left(\left(\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu\right)^{2/p}-\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu\right)\\\
\geq\lim_{t\to+\infty}\int_{{\mathbb{R}}^{d}}{|\nabla
w(t,x)^{1/p}|^{2}}\,d\mu-\frac{\kappa}{p-2}\left(\left(\int_{{\mathbb{R}}^{d}}{w(t,x)^{2/p}}\,d\mu\right)^{2/p}-M^{2/p}\right)=0\,.$

The proof relies on a commutation of the derivatives which is is similar to
the Bochner-Lichnerowicz-Weitzenböck formula (1.12). Performing a few
integrations by parts, we compute for some function
$h\in\mathrm{H}^{2}({\mathbb{R}}^{d},d\mu)$ the integral

$\displaystyle\int_{{\mathbb{R}}^{d}}{(\mathcal{L}h)^{2}}\,d\mu$
$\displaystyle=-\int_{{\mathbb{R}}^{d}}{\nabla
h\cdot\nabla(\mathcal{L}h)}\,d\mu$
$\displaystyle=-\int_{{\mathbb{R}}^{d}}{\nabla h\cdot(\mathcal{L}\nabla
h)}\,d\mu+\int_{{\mathbb{R}}^{d}}{\nabla h\cdot[\mathcal{L},\nabla]\,h}\,d\mu$
$\displaystyle=\int_{{\mathbb{R}}^{d}}{|\mathrm{Hess}h|^{2}}\,d\mu+\int_{{\mathbb{R}}^{d}}{\nabla
h\cdot[\mathcal{L},\nabla]\,h}\,d\mu\,.$

Since $\nabla h\cdot[\mathcal{L},\nabla]\,h=\mathrm{Hess}\,\phi:\nabla
h\otimes\nabla h\geq\kappa\,|\nabla h|^{2}$ according to (1.21), this proves

$\int_{{\mathbb{R}}^{d}}{(\mathcal{L}h)^{2}}\,d\mu\geq\int_{{\mathbb{R}}^{d}}{|\mathrm{Hess}h|^{2}}\,d\mu+\kappa\int_{{\mathbb{R}}^{d}}{|\nabla
h|^{2}}\,d\mu\,.$ (1.23)

Another elementary identity is given by an integration by parts:

$\int_{{\mathbb{R}}^{d}}{(\mathcal{L}h)\,\frac{|\nabla
h|^{2}}{h}}\,d\mu=\int_{{\mathbb{R}}^{d}}{\frac{|\nabla
h|^{4}}{h^{2}}}\,d\mu-2\int_{{\mathbb{R}}^{d}}{\mathrm{Hess}h:\frac{\nabla
h\otimes\nabla h}{h}}\,d\mu\,.$ (1.24)

Hence with $h=w^{1/p}$ and $w$ solving (1.20), so that $h$ solves

$\frac{\partial h}{\partial t}=\mathcal{L}h+(p-1)\,\frac{|\nabla h|^{2}}{h}$

with initial datum $h(t=0,\cdot)=|f|$, we find that

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{w^{2/p}}\,d\mu=\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{h^{2}}\,d\mu=-\,2\,(2-p)\int_{{\mathbb{R}}^{d}}{|\nabla
h|^{2}}\,d\mu\,,$ $\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{\left|\nabla
w^{1/p}\right|^{2}}\,d\mu=\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{\left|\nabla
h\right|^{2}}\,d\mu=-\,2\int_{{\mathbb{R}}^{d}}{(\mathcal{L}h)\left(\mathcal{L}h+(p-1)\,\frac{|\nabla
h|^{2}}{h}\right)}\,d\mu$

Altogether,

$\frac{d}{dt}\left(\int_{{\mathbb{R}}^{d}}{\left|\nabla
w^{1/p}\right|^{2}}\,d\mu+\frac{\kappa}{p-2}\int_{{\mathbb{R}}^{d}}{w^{2/p}}\,d\mu\right)\\\
=-\,2\int_{{\mathbb{R}}^{d}}{\left|\mathrm{Hess}h-(p-1)\,\frac{\nabla
h\otimes\nabla
h}{h}\right|^{2}}\,d\mu-\,2\,(p-1)\,(2-p)\int_{{\mathbb{R}}^{d}}{\frac{|\nabla
h|^{4}}{h^{2}}}\,d\mu\leq 0\,,$

which completes the proof of (1.22). We refer to [35, 59, 57] for further
details.

Condition (1.21) is the standard criterion in the _carré du champ_ method of
D. Bakry and M. Emery, but it is a pointwise condition which is by far too
strong, as we use it only after an integration. See [59] for some
considerations on non-local, spectral conditions. There is actually a deeper
property, for the following reason. Let us assume that $\kappa_{1}>0$ is the
optimal constant in

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\geq\frac{\kappa_{1}}{p-2}\left(\left(\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu\right)^{2/p}-\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)$
(1.25)

if $p\in[1,2)$, and in

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\geq\frac{\kappa_{1}}{2}\int_{{\mathbb{R}}^{d}}{|f|^{2}\,\log\left(|f|^{2}/M\right)}\,d\mu\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)$
(1.26)

if $p=2$, with $M=\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu$. Now, let us assume
that $\kappa_{2}>0$ is the optimal constant in

$\int_{{\mathbb{R}}^{d}}{\left|\mathrm{Hess}f-(p-1)\,\frac{\nabla
f\otimes\nabla
f}{f}\right|^{2}}\,d\mu+(p-1)\,(2-p)\int_{{\mathbb{R}}^{d}}{\frac{|\nabla
f|^{4}}{f^{2}}}\,d\mu\\\ +\int_{{\mathbb{R}}^{d}}{\mathrm{Hess}\,\phi:\nabla
f\otimes\nabla f}\,d\mu\geq\kappa_{2}\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\quad\forall\,f\in\mathrm{H}^{2}({\mathbb{R}}^{d},d\mu)\,.$
(1.27)

Finally, let us consider the Poincaré inequality

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\geq\kappa_{0}\left(\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu-\left(\int_{{\mathbb{R}}^{d}}{f}\,d\mu\right)^{2}\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)$
(1.28)

and assume that $\kappa_{0}>0$ is the optimal constant.

######  Proposition 1.3.

Let $d\geq 1$, $p\in[1,2)$, $e^{-\phi}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$,
and assume that the probability measure $d\mu=Z^{-1}\,e^{-\phi}\,dx$ with
$Z=\int_{{\mathbb{R}}^{d}}{e^{-\phi}}\,dx$ is such that (1.28) holds for some
$\kappa_{0}>0$. With the above notations, we have $\kappa_{1}>0$ and
$\kappa_{2}\leq\kappa_{1}\leq\kappa_{0}$. Moreover, if
$\kappa_{2}=\kappa_{1}$, then $\kappa_{2}=\kappa_{1}=\kappa_{0}$.

###### Proof.

By Hölder’s inequality, we know that
$\left(\int_{{\mathbb{R}}^{d}}{|f|}\,d\mu\right)^{2}\leq\left(\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu\right)^{2/p}$,
so that $\kappa_{1}\geq\kappa_{0}\,(2-p)>0$.

The fact that $\kappa_{2}\leq\kappa_{1}$ is part of the standard carré du
champ method and has already been observed above: we know that (1.25) and
(1.26) are consequences of (1.27). Inequality (1.25) or (1.26) applied to the
function $h=1+\varepsilon\,f$ gives, at leading order in the limit as
$\varepsilon\to 0$, the Poincaré inequality

$\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\geq\kappa_{1}\left(\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu-\left(\int_{{\mathbb{R}}^{d}}{f}\,d\mu\right)^{2}\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)\,,$

where the constant $\kappa_{1}$ is not necessarily optimal:
$\kappa_{1}\leq\kappa_{0}$.

If $p=1$, we have $\kappa_{2}=\kappa_{1}=\kappa_{0}$ which easily follows from
an expansion of the square: let $f\in\mathrm{H}^{1}(d\mu)$ be such that
$\int_{{\mathbb{R}}^{d}}{f}\,d\mu=0$. Using the fact that
$\int_{{\mathbb{R}}^{d}}{(\mathcal{L}f+\kappa_{0}\,f)^{2}}\,d\mu\geq 0$, we
find that

$\int_{{\mathbb{R}}^{d}}{\left|\mathrm{Hess}f\right|^{2}}\,d\mu+\int_{{\mathbb{R}}^{d}}{\mathrm{Hess}\,\phi:\nabla
f\otimes\nabla f}\,d\mu-\kappa_{0}\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu\\\ \geq\kappa_{0}\left(\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu-\kappa_{0}\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu\right)\geq
0\,,$

which proves that $\kappa_{2}\geq\kappa_{0}$. In the remainder of the proof,
we assume that $p\in(1,2)$.

In the equality case $\kappa_{2}=\kappa_{1}$, we read from the previous
computations that there is no non-constant optimal function for (1.25). This
fact is easily obtained as follows: if $f$ is an optimal function for (1.25),
let us take $w_{0}=|f|^{p}$ and consider the solution $w$ of (1.20) with
initial datum $w_{0}$. At $t=0$, we obtain
$\int_{{\mathbb{R}}^{d}}{f^{-2}\,|\nabla f|^{4}}\,d\mu=0$, a contradiction.
Hence, if we consider a minimizing sequence $(f_{n})_{n\in{\mathbb{N}}}$ for

$\kappa_{1}=\inf\mathcal{Q}_{p}[f]\quad\mbox{where}\quad\mathcal{Q}_{p}[f]:=\frac{(2-p)\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,d\mu}{\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu-\left(\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu\right)^{2/p}}\,,$

we can write that $f_{n}=1+\varepsilon_{n}\,h_{n}$ for any $n\in{\mathbb{N}}$,
with $\lim_{n\to+\infty}\varepsilon_{n}=0$ and $(h_{n})_{n\in{\mathbb{N}}}$
uniformly bounded in $\mathrm{H}^{1}({\mathbb{R}}^{d}\,d\mu)$, so that

$\kappa_{0}\geq\kappa_{1}=\lim_{n\to+\infty}\mathcal{Q}_{p}[1+\varepsilon_{n}\,h_{n}]=\lim_{n\to+\infty}\mathcal{Q}_{1}[h_{n}]\geq\kappa_{0}\,.$

For further details in a similar problem, see [20]. ∎

It is possible to prove various other estimates. For instance, the function

$p\mapsto\frac{2\,p}{2-p}\left(\int_{{\mathbb{R}}^{d}}{|f|^{2}}\,d\mu-\left(\int_{{\mathbb{R}}^{d}}{|f|^{p}}\,d\mu\right)^{2/p}\right)$

is non-decreasing according to [75, Lemma 1], so that we have the bounds

$(2-p)\,\kappa_{0}\leq\kappa_{1}\leq p\,\kappa_{0}\,.$

If the logarithmic Sobolev inequality (1.26) holds for some $\kappa_{1}>0$,
then the Poincaré inequality (1.28) holds for some $\kappa_{0}\geq\kappa_{1}$
and (1.25) holds for some constant which can be estimated in terms of
$\kappa_{0}$ and $\kappa_{1}$. Additionally, if $\kappa_{0}=\kappa_{1}$, then
$\kappa_{0}$ is also the optimal constant in (1.25) and does not depend on
$p\in[1,2]$: see [75, 3]. This occurs if $\phi(x)=|x|^{2}/2$: in that case, we
have $\kappa_{2}=\kappa_{1}=\kappa_{0}$, for any $p\in[1,2]$.

The _carré du champ_ method gives only a sufficient condition to establish
(1.25) or (1.26), and the constant $\kappa$ in (1.21) is then a lower estimate
for $\kappa_{2}$. In some cases, it can be proved by a direct computation that
$\kappa=\kappa_{2}$ and then _all optimal constants_ in (1.25)-(1.28) are
known: this is the case for the harmonic potential and, as we shall see next,
we have a similar picture in all cases of interest considered in this paper.
For this reason, the method provides sharp results. From the point of view of
the flow, if $w$ solves (1.20) with initial datum $w_{0}=|f|^{p}$, for some
non-constant function $f\in\mathrm{H}^{1}({\mathbb{R}}^{d},d\mu)$, then we
have

$\mathcal{Q}_{p}[f]\geq\lim_{t\to+\infty}\mathcal{Q}_{p}\left[w(t,\cdot)^{1/p}\right]=\lim_{t\to+\infty}\mathcal{Q}_{1}\left[w(t,\cdot)^{1/p}\right]\geq\kappa_{0}\,.$

Then $\kappa_{1}=\inf\mathcal{Q}_{p}[f]=\kappa_{0}$ is recovered using
$f=1+\varepsilon\,h$ as a test function and taking the limit as
$\varepsilon\to 0$, where $h$ is optimal for (1.28).

####  1.3.2. Flows and interpolation inequalities on the sphere

On the sphere $\mathbb{S}^{d}$, exactly the same method as for the Ornstein-
Uhlenbeck equation (1.20) applies. If we consider the heat equation

$\frac{\partial w}{\partial t}=\Delta w\,,\quad w(t=0,\cdot)=f^{p}\,,$

where $\Delta$ is the Laplace-Beltrami operator or, with $h=w^{1/p}$, the
equation

$\frac{\partial h}{\partial t}=\Delta h+(p-1)\,\frac{|\nabla
h|^{2}}{h}\,,\quad h(t=0,\cdot)=f\,,$

then we find that

$\frac{d}{dt}\int_{\mathbb{S}^{d}}{h^{p}}\,d\mu=0\,,$
$\frac{d}{dt}\int_{\mathbb{S}^{d}}{h^{2}}\,d\mu=-\,2\,(2-p)\int_{\mathbb{S}^{d}}{|\nabla
h|^{2}}\,d\mu\,,$ $\frac{d}{dt}\int_{\mathbb{S}^{d}}{|\nabla
h|^{2}}\,d\mu=-\,2\int_{\mathbb{S}^{d}}{(\Delta h)\left(\Delta
h+(p-1)\,\frac{|\nabla h|^{2}}{h}\right)}\,d\mu\,.$

The strategy of the computation is similar to the case of Equation (1.20). We
replace (1.21) by the Bochner-Lichnerowicz-Weitzenböck formula (1.12) applied
to $h$. After an integration with respect to $d\mu$, this proves

$\int_{\mathbb{S}^{d}}{(\Delta
h)^{2}}\,d\mu=\int_{\mathbb{S}^{d}}{|\mathrm{Hess}h|^{2}}\,d\mu+(d-1)\int_{\mathbb{S}^{d}}{|\nabla
h|^{2}}\,d\mu\,.$ (1.29)

After a number integrations by parts, we obtain

$\frac{d}{dt}\left[\|{\nabla
h}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{\lambda}{p-2}\left(\|{h}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\|{h}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\right]\\\
=-\,2\int_{\mathbb{S}^{d}}{\left(\tfrac{d}{d-1}\,|Q_{f}|^{2}+(d-\lambda)\,|\nabla
f|^{2}+\gamma(p)\,\frac{|\nabla f|^{4}}{f^{2}}\right)}\,d\mu$

with $Q_{f}$ and $\gamma(p)$ respectively given by (1.13) and (1.14). Using
the fact that $h^{p}\to M$ as $t\to+\infty$, we find that $\|{\nabla
h}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{\lambda}{p-2}\,\big{(}\|{h}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\|{h}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\big{)}$
written with $\lambda=d$ is monotone non-increasing, with limit $0$ as
$t\to+\infty$, so that (1.10) holds for the initial datum $h(t=0,\cdot)=f$.
This method can be found in [8] but we refer also to [53] for details on the
computations using the ultraspherical operator and to [49, 53] for some
improvements. It is however limited to $p\leq 2^{\\#}$.

It is possible to overcome the limitation $p\leq 2^{\\#}$ by considering the
nonlinear diffusion

$\frac{\partial w}{\partial t}=\Delta w^{m}\,,\quad w(t=0,\cdot)=f^{p}\,,$
(1.30)

for some appropriately chosen $m$. The idea was introduced in [43] and
systematized in [50, 49, 53]. Let us give some details on how to use (1.30).
We assume that

$p\in[1,2)\cup(2,2^{*}]\quad\mbox{if}\quad d\geq 3\quad\mbox{and}\quad
p\in[1,2)\cup(2,+\infty)\quad\mbox{if}\quad d=1,\,2\,.$ (1.31)

a range which covers the case $2^{\\#}<p<2^{*}$ and also $p=2^{*}$ if $d\geq
3$. It is convenient to replace the solution $w$ of (1.30) by $u$ such that
$w=u^{\beta\,p}$ with

$\beta=\frac{2}{2-p\,(1-m)}$

so that $u$ solves the nonlinear diffusion equation

$\frac{\partial u}{\partial t}=u^{2-2\beta}\left(\Delta
u+\kappa\,\frac{|\nabla u|^{2}}{u}\right)\,,\quad
u(t=0,\cdot)=|f|^{1/\beta}\,,$ (1.32)

with $\kappa=\beta\,(p-2)+1$. Then $h=w^{1/p}=u^{\beta}$ is such that

$\frac{d}{dt}\left[\|{\nabla
h}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{d}{p-2}\left(\|{h}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\|{h}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\right]\\\
=-\,2\,\beta^{2}\int_{\mathbb{S}^{d}}{\left(\tfrac{d}{d-1}\,|Q_{u}|^{2}+\delta(\beta)\,\frac{|\nabla
u|^{4}}{u^{2}}\right)}\,d\mu$

with $Q_{u}:=\mathrm{Hess}u-\frac{1}{d}\,\Delta
u\,\mathrm{Id}-\beta\,(p-1)\,\frac{d-1}{d+2}\left(\frac{\nabla u\otimes\nabla
u}{u}-\frac{1}{d}\,\frac{|\nabla u|^{2}}{u}\,\mathrm{Id}\right)$ and

$\delta(\beta):=-\left(\frac{d-1}{d+2}\,(\kappa+\beta-1)\right)^{2}+\,\kappa\,(\beta-1)+\,\frac{d}{d+2}\,(\kappa+\beta-1)\,.$
(1.33)

As a consequence, we can write an improved version of (1.10), with an integral
remainder term

$\|{\nabla
f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{d}{p-2}\left(\|{f}\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\|{f}\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\\\
\geq
2\,\beta^{2}\int_{0}^{+\infty}\int_{\mathbb{S}^{d}}{\left(\tfrac{d}{d-1}\,|Q_{u}|^{2}+\delta(\beta)\,\frac{|\nabla
u|^{4}}{u^{2}}\right)}\,d\mu\,dt\,.$

We recall that the optimal constant cannot be larger than $d$, as shown by a
Taylor expansion around $f=1$ of (1.10) and by the Poincaré inequality on
$\mathbb{S}^{d}$ with optimal constant, which corresponds to the case $p=1$ in
(1.10). This explains why, along the flow (1.30), optimality in (1.10) is
achieved only in the limit as $t\to+\infty$. The infimum of the quotient

$\mathcal{Q}_{p}[h]:=\frac{(2-p)\int_{\mathbb{S}^{d}}{|\nabla
h|^{2}}\,d\mu}{\int_{\mathbb{S}^{d}}{|h|^{2}}\,d\mu-\left(\int_{\mathbb{S}^{d}}{|h|^{p}}\,d\mu\right)^{2/p}}$

is obtained by considering $\lim_{\varepsilon\to
0}\mathcal{Q}_{p}[1+\varepsilon\,h]=\mathcal{Q}_{1}[h]$.

Also notice that
$\int_{\mathbb{S}^{d}}{\left(\tfrac{d}{d-1}\,|Q_{u}|^{2}+\delta(\beta)\,\frac{|\nabla
u|^{4}}{u^{2}}\right)}\,d\mu$ written for $u=1+\varepsilon\,h$ with $h$ of
zero average gives, at order $\varepsilon^{2}$, the quadratic form

$h\mapsto\frac{d}{d-1}\int_{\mathbb{S}^{d}}{\left|\mathrm{Hess}h-\frac{1}{d}\,\Delta
h\,\mathrm{Id}\right|^{2}}\,d\mu=\int_{\mathbb{S}^{d}}{(\Delta h)^{2}}\,d\mu$

and that, as a consequence of an expansion of the square in
$\int_{\mathbb{S}^{d}}{(\Delta h+d\,h)^{2}}\,d\mu$ and of the Poincaré
inequality on $\mathbb{S}^{d}$,

$\int_{\mathbb{S}^{d}}{(\Delta h)^{2}}\,d\mu\geq
d\int_{\mathbb{S}^{d}}{|\nabla h|^{2}}\,d\mu\,.$

The computation is therefore optimal for the solutions of (1.32) in the
asymptotic regime as $t\to+\infty$. In the nonlinear regime, the constant
which appears in the _carré du champ_ computation is also $d$, which
guarantees that the method gives the optimal constant in (1.10). In the words
of Section 1.3.1 and with a straightforward analogy with Proposition 1.3, we
have $\kappa_{2}=\kappa_{1}=\kappa_{0}$. This feature will again appear in
Sections 2 and 3. With the _carré du champ_ method and the flow (1.32), not
only the results of [14] are recovered in a natural parabolic setting, but
improvements like (1.15) and stability results, are also obtained: see [53,
48] for more considerations in this direction.

##  2\. Entropy methods and the fast diffusion equation on the Euclidean space

Most of this section is a summary of the results of [20]. The optimal constant
in the entropy-entropy production inequality is given by the spectral gap of
the linearized functional inequality. An important difference with Section
1.3.2 is that the linearization has to be done around the Barenblatt profile
and not around constant functions. However, from the heuristic point of view,
the situation is similar and the strategy is, _mutatis mutandis_ , to prove
that $\kappa_{2}=\kappa_{0}$. This explains why optimal constants are obtained
by generalized _carré du champ_ methods and why stability results can be
achieved for (1.3), which is the major result of [20].

###  2.1. Rényi entropy powers

On ${\mathbb{R}}^{d}$, $d\geq 1$, let us consider the _fast diffusion_
equation

$\frac{\partial u}{\partial t}=\Delta u^{m}$ (2.1)

with $m\in[m_{1},1)$, $m_{1}:=(d-1)/d$, and an initial datum
$u(t=0,x)=u_{0}(x)\geq 0$, with
$u_{0}\in\mathrm{L}^{1}\big{(}\mathbb{R}^{d},(1+|x|^{2})\,dx\big{)}$. Equation
(2.1) was interpreted as the gradient flow of the _entropy_

$\mathsf{E}:=\int_{{\mathbb{R}}^{d}}{u^{m}}\,dx$

with respect to Wasserstein’s distance in [85]. Equation (2.1) admits self-
similar solutions, the so-called _Barenblatt functions_ which, up to a
scaling, a multiplication by a constant and a translation, take the form

$B(t,x)=\frac{1}{\kappa\,R(t)^{d}}\,\mathcal{B}\\!\left(\tau(t),\frac{x}{\kappa\,R(t)}\right)\quad\forall\,(t,x)\in{\mathbb{R}}^{+}\times{\mathbb{R}}^{d}$

for some numerical constant $\kappa>0$, an unbounded increasing function
$t\mapsto R(t)$, $\tau(t)=\frac{1}{2}\,\log R(t)$ and a Barenblatt profile
$\mathcal{B}$ that can be written as

$\mathcal{B}(x)=\left(1+|x|^{2}\right)^{\frac{1}{m-1}}\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

As in Section 1.3.2, we prove (1.3) directly by the _carré du champ method_.
The key property, inspired by the _Rényi entropy powers_ of [88] is based on:
(i) The entropy growth estimate:

$\mathsf{E}^{\prime}=(1-m)\,\mathsf{I}\quad\mbox{where}\quad\mathsf{I}:=\int_{{\mathbb{R}}^{d}}{u\,\left|\nabla\mathsf{P}\right|^{2}}\,dx\quad\mbox{and}\quad\mathsf{P}:=\frac{m}{m-1}\,u^{m-1}\,.$

Here $\mathsf{I}$ and $\mathsf{P}$ are known respectively as the generalized
_Fisher information_ and the _pressure variable_.
(ii) The $t$-derivative of the generalized _Rényi entropy powers_ , that is,

$\mathsf{G}:=\mathsf{I}\,\mathsf{E}^{\,2\,\frac{m-m_{1}}{1-m}}\,,$

satisfies the identity

$-\frac{1}{2}\,\frac{d}{dt}\log\mathsf{G}=\int_{{\mathbb{R}}^{d}}{u^{m}\,\left|\,\mathrm{D}^{2}\mathsf{P}-\frac{1}{d}\,\Delta\mathsf{P}\,\mathrm{Id}\,\right|^{2}}\,dx+(m-m_{1})\int_{{\mathbb{R}}^{d}}{u^{m}\,\left|\,\Delta\mathsf{P}+\frac{\mathsf{I}}{\mathsf{E}}\,\right|^{2}}\,dx\,.$
(2.2)

Hence $\mathsf{G}$ is monotone with a limit given by a self-similar
_Barenblatt function_ $B(t,x)$. It turns out that writing
$\mathsf{G}[u]\geq\mathsf{G}[B]$ is exactly (1.3) written with the optimal
constant, using the relations $f=u^{m-1/2}$, so that, with

$p=\frac{1}{2\,m-1}\,,$

we have $\int_{\mathbb{R}^{d}}u\,dx=\int_{\mathbb{R}^{d}}f^{2p}\,dx$,
$\mathsf{E}=\int_{\mathbb{R}^{d}}f^{p+1}\,dx$ and
$\mathsf{I}=(p+1)^{2}\int_{\mathbb{R}^{d}}|\nabla f|^{2}\,dx$. Inequality
(1.3) is then easily recovered.

This gives the growth rate of $\mathsf{E}$. Let

$C_{0}:=4\,\tfrac{(1-m)^{3}}{(2\,m-1)^{2}}\,\left(\mathcal{C}_{\mathrm{GNS}}(p)\right)^{\frac{2}{d}\frac{(d+2)\,m-d}{(1-m)\,(2\,m-1)}}\,M^{\frac{(d+2)\,m-d}{d\,(1-m)}}$

where $M=\|{u_{0}}\|_{1}$, and $m_{c}:=(d-2)/d$. According to [20, Lemma 2.1],
we have the following result.

######  Lemma 2.1 ([20]).

Assume that $d\geq 1$, $m\in[m_{1},1)$ with additionally $m>1/2$ if $d=1$ or
$d=2$, and consider a solution of (2.1) with initial datum
$u_{0}\in\mathrm{L}^{1}_{+}\\!\left({\mathbb{R}}^{d},(1+|x|^{2})\,dx\right)$
such that $u_{0}^{m}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$. Then

$\int_{{\mathbb{R}}^{d}}{u^{m}(t,x)}\,dx\geq\left(\int_{{\mathbb{R}}^{d}}{u_{0}^{m}}\,dx+\tfrac{(1-m)\,C_{0}}{m-m_{c}}\,t\right)^{\frac{1-m}{m-m_{c}}}\quad\forall\,t\geq
0\,.$ (2.3)

Knowing the optimal constant $C_{0}$ in (2.3) is equivalent to identifying the
optimal constant in (1.3). This is a standard property of entropy methods that
optimal decay rates in the evolution equation are equivalent to optimal
constants in the corresponding functional inequalities. The threshold case
$m=m_{1}$ in (2.1) corresponds to $p=p_{\star}$ in (1.3), _i.e._ , to the
Sobolev inequality.

###  2.2. Relative entropy and relative Fisher information

Proving (2.2) requires some integrations by parts which are delicate to
justify in the class of solutions considered in Lemma 2.1. A rigorous proof
goes through the time-dependent rescaling

$u(t,x)=\frac{1}{\kappa^{d}\,R^{d}}\,v\\!\left(\tau,\frac{x}{\kappa\,R}\right)\quad\mbox{where}\quad\frac{dR}{dt}=R^{-d\,(m-m_{c})}\,,\quad\tau(t):=\tfrac{1}{2}\,\log
R(t)\,,$ (2.4)

so that (2.1) is changed, with the choice $R(0)=1$, into the _Fokker-Planck
type equation_

$\frac{\partial v}{\partial\tau}+\nabla\cdot\Big{[}v\left(\nabla
v^{m-1}-\,2\,x\right)\Big{]}=0\,,$ (2.5)

with an initial datum $v_{0}:=\kappa^{d}\,u_{0}(\kappa\cdot)\geq 0$, for some
numerical parameter $\kappa$ which depends only on $m$. For simplicity, we
shall assume from now on that
$\int_{{\mathbb{R}}^{d}}{v_{0}}\,dx=\int_{{\mathbb{R}}^{d}}{\mathcal{B}}\,dx$.

For a given function
$v\in\mathrm{L}^{1}_{+}\\!\left({\mathbb{R}}^{d},(1+|x|^{2})\,dx\right)$ such
that $v^{m}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$ and
$\int_{{\mathbb{R}}^{d}}{v}\,dx=\int_{{\mathbb{R}}^{d}}{\mathcal{B}}\,dx$, let
us define the _relative entropy_ (or _free energy_) and the _relative Fisher
information_ respectively by

$\mathcal{F}[v]\\!:=\\!\int_{\mathbb{R}^{d}}\left(\mathcal{B}^{m-1}\,(v-\mathcal{B})-\tfrac{1}{m}\,(v^{m}-\mathcal{B}^{m})\right)dx\,,\quad\mathcal{I}[v]\\!:=\\!\int_{\mathbb{R}^{d}}v\left|\nabla
v^{m-1}+\,2\,x\right|^{2}\,dx\,.$

It is proved in [41] that the _entropy-entropy production inequality_

$\mathcal{I}[v]\geq 4\,\mathcal{F}[v]$ (2.6)

is exactly equivalent to (1.3) written with the optimal constant. This is not
difficult to check using $f=v^{m-1/2}$ and $\mathcal{B}=\mathsf{g}^{2p}$ with
$p=1/(2\,m-1)$.

If additionally $v$ solves (2.5), it is almost straightforward to check that

$\frac{d}{dt}\mathcal{F}[v(t,\cdot)]=-\,\mathcal{I}[v(t,\cdot)]\leq-\,4\,\mathcal{F}[v(t,\cdot)]\,,$
(2.7)

so that we have the estimate

$\mathcal{F}[v(t,\cdot)]\leq\mathcal{F}[v_{0}]\,e^{-4t}\quad\forall\,t\geq
0\,.$

This can be used to obtain intermediate asymptotics for a solution of (2.1)
using (2.4) and a generalized Pinsker-Csiszár-Kullback inequality. Even more
interesting is the fact that the _carré du champ_ method applies:

$\frac{d}{dt}\mathcal{I}[v(t,\cdot)]\leq-\,4\,\mathcal{I}[v(t,\cdot)]\,,$
(2.8)

with equality up to a term which is similar to the one in (2.2) after applying
(2.4). The justification of the integrations by parts is easier than in the
Rényi entropy powers approach, but still requires some care: see [71, 51] and
references therein for a complete proof. This is also the right setting to
prove the identities in the Rényi entropy framework of Section 2.1, up to the
change of variables (2.4): all details can be found in [51].

Let us consider the quotient

$\mathcal{Q}[v]:=\frac{\mathcal{I}[v]}{\mathcal{F}[v]}\,.$

Based on (2.7) and (2.8), we obtain that

$\frac{d}{dt}\mathcal{Q}[v(t,\cdot)]=\mathcal{Q}[v(t,\cdot)]\,\big{(}\mathcal{Q}[v(t,\cdot)]-4\big{)}$

if $v$ solves (2.5). This proves the following _initial time layer_ property.

######  Lemma 2.2 ([20]).

With the above assumptions, if $\mathcal{Q}[v(T,\cdot)]\geq 4+\eta$ for some
$\eta>0$ and $T>0$, then
$\mathcal{Q}[v(t,\cdot)]\geq\,4+4\,\eta\,e^{-4\,T}/(4+\eta-\eta\,e^{-4\,T})$
for any $t\in[0,T]$.

###  2.3. Linearization and Hardy-Poincaré inequalities

For reasons which are similar to the ones in Section 1.3.2, it turns out that
$\inf\mathcal{Q}[v]=4$ is not achieved among admissible functions, in the
sense that $\lim_{n\to+\infty}\mathcal{F}[v_{n}]=0$ if
$(v_{n})_{n\in{\mathbb{N}}}$ is a minimizing sequence for $\mathcal{Q}$. In
other words and under appropriate normalization conditions, this means that
$v_{n}\to\mathcal{B}$ and suggests to consider the limiting problem

$\inf\frac{\mathsf{I}[h]}{\mathsf{F}[h]}=\lim_{\varepsilon\to
0}\mathcal{Q}\big{[}\mathcal{B}\,\big{(}1+\varepsilon\,\mathcal{B}^{1-m}\,h\big{)}\big{]}$

where the _linearized free energy_ and the _linearized Fisher information_ are
defined respectively by

$\mathsf{F}[h]:=\frac{m}{2}\int_{{\mathbb{R}}^{d}}{|h|^{2}\,\mathcal{B}^{2-m}}\,dx\quad\mbox{and}\quad\mathsf{I}[h]:=m\,(1-m)\int_{{\mathbb{R}}^{d}}{|\nabla
h|^{2}\,\mathcal{B}}\,dx\,.$

By the _Hardy-Poincaré inequality_ if $d\geq 1$ and $m\in(m_{1},1)$, and for
any function $h\in\mathrm{L}^{2}({\mathbb{R}}^{d},\mathcal{B}^{2-m}\,dx)$ such
that $\nabla h\in\mathrm{L}^{2}({\mathbb{R}}^{d},\mathcal{B}\,dx)$ and
$\int_{{\mathbb{R}}^{d}}{h\,\mathcal{B}^{2-m}}\,dx=0$, we have the _Hardy-
Poincaré inequality_

$\mathsf{I}[h]\geq 4\,\mathsf{F}[h]\,.$

If additionally we assume that
$\int_{{\mathbb{R}}^{d}}{x\,h\,\mathcal{B}^{2-m}}\,dx=0$, then we have the
_improved Hardy-Poincaré inequality_

$\mathsf{I}[h]\geq 4\,\alpha\,\mathsf{F}[h]$ (2.9)

where $\alpha=2-d\,(1-m)$ if $m\in(m_{1},1)$. To get an improved inequality if
$m=m_{1}$, one has to assume additionally that
$\int_{{\mathbb{R}}^{d}}{|x|^{2}\,h\,\mathcal{B}^{2-m}}\,dx=0$. As in [15, 17,
20], the improved estimate of (2.9) can be reimported in the nonlinear
functionals as follows.

######  Lemma 2.3 ([20]).

Let $m\in(m_{1},1)$ if $d\geq 2$, $m\in(1/3,1)$ if $d=1$,
$\eta=2\,d\,(m-m_{1})$. There exists an explicit $\chi\in(0,1)$ such that,if
$\int_{{\mathbb{R}}^{d}}{v}\,dx=\mathcal{M}$,
$\int_{{\mathbb{R}}^{d}}{x\,v}\,dx=0$ and

$(1-\varepsilon)\,\mathcal{B}\leq v\leq(1+\varepsilon)\,\mathcal{B}$ (2.10)

for some $\varepsilon\in(0,\chi\,\eta)$, then

$\mathcal{Q}[v]\geq 4+\eta\,.$

In other words, since Condition (2.10) is stable under the action of (2.5)
according to [15], we have the _asymptotic time layer_ property
$\mathcal{Q}[v(t,\cdot)]\geq 4+\eta$ if the solution $v(t,\cdot)$ of (2.5)
satisfies (2.10) for any $t>T_{\star}$, for some _threshold time_
$T_{\star}>0$.

###  2.4. The threshold time estimate

The existence of a _threshold time_ is known for a given initial datum from
[22], but has been quantified only in [20] as follows.

######  Theorem 2.4 ([20]).

Let $m\in\big{[}m_{1},1\big{)}$ if $d\geq 2$, $m>1/3$ if $d=1$, $A>0$, and
$G>0$ be given. Let $\varepsilon\in(0,\min\\{\chi\,\eta,\varepsilon_{0}\\})$,
with $\eta$ and $\chi$ as in Proposition 2.3, for some explicit
$\varepsilon_{0}\in(0,1/2)$, and define

$T_{\star}:=\frac{1}{2\,\alpha}\,\log\left(1+\alpha\,{\mathsf{c}_{\star}}\,\frac{1+A^{1-m}+G^{\frac{\alpha}{2}}}{\varepsilon^{\mathsf{a}}}\right)$
(2.11)

with $\alpha=d\,(m-m_{c})$, for some explicit, positive numerical constants
${\mathsf{c}_{\star}}>0$ and $\mathsf{a}$. Then for any solution $v$ to (2.5)
with nonnegative initial datum $v_{0}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$,
$\int_{{\mathbb{R}}^{d}}{v_{0}}\,dx=\mathcal{M}$,
$\int_{{\mathbb{R}}^{d}}{x\,v_{0}}\,dx=0$ which satisfies

$\sup_{r>0}r^{\frac{d\,(m-m_{c})}{(1-m)}}\int_{|x|>r}u_{0}\,dx\leq
A\,,\quad\mathcal{F}[v_{0}]\leq G\,,$ (2.12)

we have that

$(1-\varepsilon)\,\mathcal{B}(x)\leq
v(t,x)\leq(1+\varepsilon)\,\mathcal{B}(x)\quad\forall\,(t,x)\in[T_{\star},+\infty)\times{\mathbb{R}}^{d}\,.$
(2.13)

Based on a _global Harnack Principle_ , this result is at the core of the
method of [20]. It relies on a quantitative version of the results of J. Moser
in [79, 80], and a constructive proof of [22] based on the improved results of
[21]. As discussed in [20, Chapter 7], the tail decay in (2.12) is not only
sufficient but also necessary for obtaining (2.13).

###  2.5. A stability result for Gagliardo-Nirenberg-Sobolev inequalities

By collecting the _initial time layer_ property of of Lemma 2.2, the
_asympotic time layer_ property of Lemma 2.3 and the _threshold time estimate_
of Theorem 2.4, we are able to write an improved entropy-entropy production
inequality which amounts to
$\mathcal{Q}[v(t,\cdot)]\geq\,4+4\,\eta\,e^{-4\,T_{\star}}/(4+\eta-\eta\,e^{-4\,T_{\star}})$
for any $t\in{\mathbb{R}}^{+}$. This is the desired stability result, which
goes as follows. There is a unique $\mathsf{g}_{f}\in\mathfrak{M}$ such that

$\int_{{\mathbb{R}}^{d}}{(1,x,|x|^{2})\,f^{2p}}\,dx=\int_{{\mathbb{R}}^{d}}{(1,x,|x|^{2})\,\mathsf{g}_{f}^{2p}}\,dx\,.$

By the entropy-entropy production inequality (2.6) written for $v=f^{2p}$, we
know that

$\delta[f]:=\frac{p+1}{p-1}\int_{{\mathbb{R}}^{d}}{\left|(p-1)\,\nabla
f+f^{p}\,\nabla\mathsf{g}_{f}^{1-p}\right|^{2}}\,dx-\,4\,\mathcal{E}[f]$

is nonnegative, where the _relative entropy_ with respect to $\mathsf{g}_{f}$
is defined by

$\mathcal{E}[f]:=\frac{2\,p}{1-p}\int_{{\mathbb{R}}^{d}}{\left(f^{p+1}-\mathsf{g}_{f}^{p+1}-\tfrac{1+p}{2\,p}\,\mathsf{g}_{f}^{1-p}\left(f^{2p}-\mathsf{g}_{f}^{2p}\right)\right)}\,dx\,.$

An expansion of $\delta$ shows that, for some explicit numerical constant
$\mathcal{K}_{\mathrm{GNS}}$,

$\delta[f]=(p-1)^{2}\,\|{\nabla
f}\|_{2}^{2}+4\,\frac{d-p\,(d-2)}{p-1}\,\|{f}\|_{p+1}^{p+1}-\mathcal{K}_{\mathrm{GNS}}\,\frac{p+1}{p-1}\,\|{f}\|_{2p}^{2p\gamma}$

and the inequality $\delta[f]\geq 0$ is actually equivalent to (1.3).

######  Theorem 2.5 (Stability in Gagliardo-Nirenberg inequalities, [20]).

Let $d\geq 1$ and $p\in(1,p^{\star})$. There is an explicit $\mathcal{C}$ such
that, for any nonnegative
$f\in\mathrm{L}^{2p}\big{(}\mathbb{R}^{d},(1+|x|^{2})\,dx\big{)}$ such that
$\nabla f\in\mathrm{L}^{2}(\mathbb{R}^{d})$ and

$A=\sup_{r>0}r^{\frac{d-p\,(d-4)}{p-1}}\int_{|x|>r}f^{2p}\,dx<\infty\quad\mbox{and}\quad\mathcal{E}[f]=G<\infty\,,$

then we have

$\delta[f]\geq\mathcal{C}\inf_{\varphi\in\mathfrak{M}}\int_{{\mathbb{R}}^{d}}{\left|(p-1)\,\nabla
f+f^{p}\,\nabla\varphi^{1-p}\right|^{2}}\,dx\,.$

The main point is that $\mathcal{C}$ has an explicit expression in terms of
$A$ and $G$, which does not degenerate to zero if $f\in\mathfrak{M}$. The
critical case $p=p^{\star}$ can also be covered up to an additional scaling:
see [20, Chapter 6].

######  Theorem 2.6 (Stability in Sobolev inequalities, [20]).

Let $d\geq 3$ and $A>0$.There is an explicit $\mathcal{C}$ such that, for any
nonnegative function
$f\in\mathrm{L}^{2^{*}}\big{(}\mathbb{R}^{d},(1+|x|^{2})\,dx\big{)}$ such that
$\nabla f\in\mathrm{L}^{2}(\mathbb{R}^{d})$ and

$\int_{{\mathbb{R}}^{d}}{(1,x,|x|^{2})\,f^{2^{*}}}\,dx=\int_{{\mathbb{R}}^{d}}{(1,x,|x|^{2})\,\mathsf{g}}\,dx\quad\mbox{and}\quad\sup_{r>0}r^{d}\int_{|x|>r}\,f^{2^{*}}\,dx\leq
A\,,$

we have

$\|{\nabla
f}\|_{2}^{2}-\mathsf{S}_{d}\,\|{f}\|_{2^{*}}^{2}\geq\mathcal{C}\int_{{\mathbb{R}}^{d}}{\left|\tfrac{2}{d-2}\,\nabla
f+f^{\frac{d}{d-2}}\,\nabla\mathsf{g}^{-\frac{2}{d-2}}\right|^{2}}\,dx\,.$
(2.14)

The stability constant is $\mathcal{C}$ in Theorem 2.6 depends only on $A$. In
Theorems 2.5 and 2.6, the stability is measured by the _relative Fisher
information_ , that is, by

$\int_{{\mathbb{R}}^{d}}{\left|(p-1)\,\nabla
f+f^{p}\,\nabla\varphi^{1-p}\right|^{2}}\,dx\,.$

Such a quantity differs from the distance to $\mathfrak{M}$ in (1.2) and
deserves some comments. A simple expansion of the square and an optimization
under scalings proves the following nonlinear extension of the _Heisenberg
uncertainty principle_ ,

$\left(\frac{d}{p+1}\int_{{\mathbb{R}}^{d}}{f^{p+1}}\,dx\right)^{2}\leq\int_{{\mathbb{R}}^{d}}{|\nabla
f|^{2}}\,dx\int_{{\mathbb{R}}^{d}}{|x|^{2}\,f^{2p}}\,dx\,.$ (2.15)

As a consequence of (1.3), _i.e._ , of $\delta[f]\geq 0$, we have that

$\int_{{\mathbb{R}}^{d}}{\left|(p-1)\,\nabla
f+f^{p}\,\nabla\varphi^{1-p}\right|^{2}}\,dx\geq
4\,\frac{p-1}{p+1}\,\mathcal{E}[f]\,.$

According to [20, Lemma 1.7], by the _Pinsker-Csiszár-Kullback inequality_ ,
$\mathcal{E}[f]$ controls a more standard distance between $f^{2p}$ and
$\mathsf{g}_{f}^{2p}$, namely

$\mathcal{E}[f]\geq\frac{p+1}{8\,p}\left(\int_{{\mathbb{R}}^{d}}{\mathsf{g}_{f}^{3p-1}}\,dx\right)^{-1}\,\left|f^{2p}-\mathsf{g}_{f}^{2p}\right|_{1}^{2}\,.$

##  3\. Symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg
inequalities

This section is devoted to Theorem 1.1. The interpretation of (1.5) as an
entropy-entropy production inequality paves the way to a flow method which
generalizes the computations done for the fast diffusion equation in Section
2. By the _carré du champ_ method, we are able to reduce the symmetry issue to
a simple dichotomy:
$\rhd$ either the entropy-entropy production inequality is linearly unstable
around the Barenblatt type profiles and optimal functions are not radially
symmetric,
$\rhd$ or the linearized entropy-entropy production inequality admits a
spectral gap which also enters in the _carré du champ_ estimates, in the
nonlinear regime, as for the fast diffusion equation, and then the inequality
holds with precisely the spectral gap as optimal constant. Using the analogy
with Section 1.3.2, we are again in a case for which $\kappa_{2}=\kappa_{0}$.
More details can be found in [51, Section 4.2].

###  3.1. Symmetry versus symmetry breaking

Let us consider the _critical Caffarelli-Kohn-Nirenberg inequalities_ (1.5).
In dimension $d\geq 2$, it has been proved in [34] that an optimal function
$f\geq 0$ exists, if $b<a+1$, and also $b>a$ if $a<0$. With the appropriate
normalization, $f$ solves

$-\mathrm{div}\big{(}\kern 0.5pt|x|^{-2a}\,\nabla f\big{)}=|x|^{-\kern
0.5ptb\kern 0.5ptp}\,f^{p-1}\,.$ (3.1)

The _symmetry versus symmetry breaking issue_ is to decide whether or not $f$
is radially symmetric. The answer is given by Theorem 1.1, but a slightly more
general _rigidity result_ is given in [52]: if $b\leq b_{\,\rm FS}(a)<b<a+1$,
up to a scaling, $\mathsf{g}$ as defined in (1.6) is the unique positive
solution of (3.1). This symmetry result cannot be handled so far with
symmetrization methods: in the range of $(a,b)$ for which

$u\mapsto\int_{{\mathbb{R}}^{d}}{\frac{|\nabla
u|^{2}}{|x|^{2\,a}}}\,dx-\frac{\|{\nabla\mathsf{g}\,|x|^{-a}}\|_{2}^{2}}{\|{\mathsf{g}\,|x|^{-b}}\|_{p}^{2}}\left(\int_{{\mathbb{R}}^{d}}{\frac{|u|^{p}}{|x|^{b\,p}}}\,dx\right)^{2/p}$

is linearly stable around $\mathsf{g}$, a local property, then $\mathsf{g}$ is
a global minimizer and, as a consequence,
$\mathsf{C}_{a,b}=\|{\nabla\mathsf{g}\,|x|^{-a}}\|_{2}^{2}/\|{\mathsf{g}\,|x|^{-b}}\|_{p}^{2}$.

###  3.2. Strategy of the proof

We summarize here the strategy of the proof in [52], which we decompose in
four steps.

####  3.2.1. A modified Sobolev inequality

Let us use spherical coordinates with $r=|x|$ and $\omega=x/r$ for any
$x\in{\mathbb{R}}^{d}\setminus\\{0\\}$. With the change of variables

$f(r,\omega)=F(s,\omega)\quad\mbox{with}\quad s=r^{\alpha}\,,$

Inequality (1.5) is transformed into a Sobolev type inequality

$\int_{{\mathbb{R}}^{d}}{|\mathsf{D}_{\alpha}\kern
1.0ptF|^{2}}\,d\mu\geq\mathsf{C}^{\kern
0.25ptd}_{a,b}\,\alpha^{1-\frac{2}{p}}\left(\int_{{\mathbb{R}}^{d}}{|F|^{p}}\,d\mu\right)^{\frac{2}{p}}$
(3.2)

where $d\mu=s^{n-1}\,ds\,d\omega=|x|^{n-d}\,dx$,

$\mathsf{D}_{\alpha}\kern 1.0ptF:=\left(\alpha\,\frac{\partial F}{\partial
s},\frac{1}{s}\,\nabla_{\omega}F\right)$

so that $|\mathsf{D}_{\alpha}\kern
1.0ptF|^{2}=\alpha^{2}\,\big{(}\frac{\partial F}{\partial
s}\big{)}^{2}+\frac{|\nabla_{\omega}F|^{2}}{s^{2}}$, and $\alpha$ and $n$ are
two parameters defined by

$\alpha=\frac{(1+a-b)\,(a_{c}-a)}{a_{c}-a+b}\,,\quad n=\frac{2\,p}{p-2}\,.$

Here $n$ plays the role of an artificial dimension while $\alpha$ reflects a
discrepancy between derivatives with respect to the radial and angular
variables. The parametrization by $a$ and $b$ of (1.5) is now replaced by the
parametrization by $\alpha>0$ and $n>d$. The Felli & Schneider curve
$b=b_{\kern 0.5pt\rm FS}(a)$ becomes $\alpha=\alpha_{\rm FS}$ in the new set
of parameters $\alpha$ and $n$, with

$\alpha_{\rm FS}:=\sqrt{\frac{d-1}{n-1}}\,.$

For $\alpha>\alpha_{\rm FS}$ the minimizers are not radial. With
$u=|F|^{p}={\mathsf{P}}^{-n}$, Inequality (3.2) becomes

$\int_{{\mathbb{R}}^{d}}{u\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}}\,d\mu\geq\frac{4\,\alpha^{1-\frac{2}{p}}}{(n-2)^{2}}\,\mathsf{C}^{\kern
0.25ptd}_{a,b}\left(\int_{{\mathbb{R}}^{d}}{u}\,d\mu\right)^{\frac{2}{p}}$
(3.3)

with $p=2\,n/(n-2)$ while (3.1) is transformed into

$-\,\mathcal{L}\,F=F^{p-1}\quad\mbox{where}\quad\mathcal{L}:=-\,\mathsf{D}_{\alpha}^{*}\kern
1.0pt\cdot\mathsf{D}_{\alpha}\kern 1.0pt\,.$ (3.4)

####  3.2.2. The flow

The fast diffusion flow

$\frac{\partial\kern 0.5ptu}{\partial
t}=\mathcal{L}\,u^{1-\frac{1}{n}}=(1-n)\,\mathsf{D}_{\alpha}^{*}\kern
1.0pt\cdot\big{(}u\,\mathsf{D}_{\alpha}\kern 1.0pt{\mathsf{P}}\big{)}$ (3.5)

is similar to the flow studied in Section 2.1 and admits self-similar
solutions

$B(t,x)=t^{-n}\left(c+\frac{|x|^{2}}{2\,(n-1)\,\alpha^{2}\,t^{2}}\right)^{-n}$

where $c>0$ can be adjusted so that $\int_{{\mathbb{R}}^{d}}{B(t,x)}\,d\mu=$
for any given $M>0$. Let us notice that $M=\int_{{\mathbb{R}}^{d}}{u}\,d\mu$
is conserved by (3.5) and formally compute

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{u\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}}\,d\mu=-\,2\int_{{\mathbb{R}}^{d}}{\left(\frac{1}{2}\,\mathcal{L}\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}-\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}\cdot\mathsf{D}_{\alpha}\kern
1.0pt\mathcal{L}\,{\mathsf{P}}-\frac{1}{n}\,(\mathcal{L}\,{\mathsf{P}})^{2}\right)}\,d\mu\,.$

Hence, if $u$ is optimal for (3.3), it is clear that
$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{u\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}}\,d\mu=0$.

####  3.2.3. The _carré du champ_ method

Let us denote by $\nabla$ the gradient with respect to the angular variable
$\omega\in{\mathbb{S}}^{d-1}$ and by $\Delta$ the Laplace-Beltrami operator on
${\mathbb{S}}^{d-1}$. The following algebraic computation is adapted from [52]
and extends to (3.5) the _carré du champ_ method. With $n>d\geq 3$ and the
notation ${}^{\prime}=\partial_{s}$, we have

$\frac{1}{2}\,\mathcal{L}\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}-\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}\cdot\mathsf{D}_{\alpha}\kern
1.0pt\mathcal{L}\,{\mathsf{P}}-\frac{1}{n}\,(\mathcal{L}\,{\mathsf{P}})^{2}\\\
=\alpha^{4}\,\frac{n-1}{n}\left({\mathsf{P}}^{\prime\prime}-\frac{{\mathsf{P}}^{\prime}}{r}-\frac{\Delta\kern
0.5pt{\mathsf{P}}}{\alpha^{2}\,(n-1)\,r^{2}}\right)^{2}+\frac{2\,\alpha^{2}}{r^{2}}\Big{|}\nabla{\mathsf{P}}^{\prime}-\frac{\nabla{\mathsf{P}}}{r}\Big{|}^{2}\\\
+\,\frac{1}{r^{4}}\left(\frac{1}{2}\,\Delta\kern
0.5pt|\nabla{\mathsf{P}}|^{2}-\nabla{\mathsf{P}}\cdot\nabla\Delta\kern
0.5pt{\mathsf{P}}-\frac{1}{n-1}(\Delta\kern
0.5pt{\mathsf{P}})^{2}-(n-2)\,\alpha^{2}\,|\nabla{\mathsf{P}}|^{2}\right)\,.$

If ${\mathsf{P}}$ is a positive function in $C^{3}(\mathbb{S}^{d-1})$ and
$d\geq 3$, the computation of Section 1.3.2 can be adapted to prove that

$\displaystyle\kern-24.0pt\int_{\mathbb{S}^{d-1}}\left(\frac{1}{2}\,\Delta\kern
0.5pt|\nabla{\mathsf{P}}|^{2}-\nabla{\mathsf{P}}\cdot\nabla\Delta\kern
0.5pt{\mathsf{P}}-\frac{1}{n-1}(\Delta\kern
0.5pt{\mathsf{P}})^{2}-(n-2)\,\alpha^{2}\,|\nabla{\mathsf{P}}|^{2}\right){\mathsf{P}}^{1-n}\,d\omega$
$\displaystyle=$
$\displaystyle\frac{(n-2)\,(d-1)}{(n-1)\,(d-2)}\int_{\mathbb{S}^{d-1}}|\mathsf{Q}|^{2}\,{\mathsf{P}}^{1-n}\,d\omega$
$\displaystyle+\,\frac{n-d}{2\,(d+1)}\left(\frac{n+3}{2}+\frac{3\,(n-1)\,(n+1)\,(d-2)}{2\,(n-2)\,(d+1)}\right)\int_{\mathbb{S}^{d-1}}{\frac{|\nabla{\mathsf{P}}|^{4}}{{\mathsf{P}}^{2}}{\mathsf{P}}^{1-n}}\,d\omega$
$\displaystyle+\,(n-2)\left(\alpha_{\rm
FS}^{2}-\alpha^{2}\right)\int_{\mathbb{S}^{d-1}}{|\nabla{\mathsf{P}}|^{2}\,{\mathsf{P}}^{1-n}}\,d\omega$

where
$\mathsf{Q}:=(\nabla\otimes\nabla)\,{\mathsf{P}}-\frac{1}{d-1}\,(\Delta\kern
0.5pt{\mathsf{P}})\,\mathrm{Id}-\frac{3\,(n-1)\,(n-d)}{2\,(n-2)\,(d+1)}\left(\frac{\nabla{\mathsf{P}}\otimes\nabla{\mathsf{P}}}{{\mathsf{P}}}-\frac{1}{d-1}\,\frac{|\nabla{\mathsf{P}}|^{2}}{{\mathsf{P}}}\,\mathrm{Id}\right)$.
For the case $d=2$ we refer the reader to [52].

######  Lemma 3.1.

Assume that $d\geq 3$. If $u$ is a smooth enough and sufficiently decaying
function as $|x|=r\to+\infty$, then

$\displaystyle\kern-24.0pt\int_{{\mathbb{R}}^{d}}{\left(\frac{1}{2}\,\mathcal{L}\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}-\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}\cdot\mathsf{D}_{\alpha}\kern
1.0pt\mathcal{L}\,{\mathsf{P}}-\frac{1}{n}\,(\mathcal{L}\,{\mathsf{P}})^{2}\right)}\,d\mu$
$\displaystyle=$
$\displaystyle\alpha^{4}\,\frac{n-1}{n}\int_{{\mathbb{R}}^{d}}{\left({\mathsf{P}}^{\prime\prime}-\frac{{\mathsf{P}}^{\prime}}{r}-\frac{\Delta\kern
0.5pt{\mathsf{P}}}{\alpha^{2}\,(n-1)\,r^{2}}\right)^{2}}\,d\mu+2\,\alpha^{2}\int_{{\mathbb{R}}^{d}}{\frac{1}{r^{2}}\Big{|}\nabla{\mathsf{P}}^{\prime}-\frac{\nabla{\mathsf{P}}}{r}\Big{|}^{2}}\,d\mu$
$\displaystyle+\,\frac{(n-2)\,(d-1)}{(n-1)\,(d-2)}\int_{{\mathbb{R}}^{d}}{\frac{1}{r^{4}}\,|\mathsf{Q}|^{2}\,{\mathsf{P}}^{1-n}}\,d\mu$
$\displaystyle+\,\frac{n-d}{2\,(d+1)}\left(\frac{n+3}{2}+\frac{3\,(n-1)\,(n+1)\,(d-2)}{2\,(n-2)\,(d+1)}\right)\int_{{\mathbb{R}}^{d}}{\frac{1}{r^{4}}\,\frac{|\nabla{\mathsf{P}}|^{4}}{{\mathsf{P}}^{2}}{\mathsf{P}}^{1-n}}\,d\mu$
$\displaystyle+\,(n-2)\left(\alpha_{\rm
FS}^{2}-\alpha^{2}\right)\int_{{\mathbb{R}}^{d}}{\frac{1}{r^{4}}|\nabla{\mathsf{P}}|^{2}\,{\mathsf{P}}^{1-n}}\,d\mu\,.$

Up to regularity issues and decay properties as $|x|\to+\infty$, we conclude
that a solution of (3.5) such that
$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{u\,|\mathsf{D}_{\alpha}\kern
1.0pt{\mathsf{P}}|^{2}}\,d\mu=0$ satisfies $\nabla{\mathsf{P}}=0$ and
${\mathsf{P}}^{\prime\prime}-{\mathsf{P}}^{\prime}/r=0$ if
$\alpha\leq\alpha_{\rm FS}$, which means that it takes the form
$u^{-n}={\mathsf{P}}=\mathsf{a}+\mathsf{b}\,r^{2}$ for some $t$-dependent
coefficients $\mathsf{a}$ and $\mathsf{b}$. Inserting this expression in
(3.5), we obtain $u=B$, up to a scaling, that is, a translation in time in
self-similar variables.

####  3.2.4. Regularity issues

The above computation is so far formal, as integrations by parts are not
justified. In [52], the regularity and decay properties needed for the method
are established for positive solutions of (3.4) by elliptic estimates. In
[51], partial results are obtained for the parabolic point of view using self-
similar variables as in Section 2.2. A complete parabolic proof is however
still missing.

##  4\. Conclusion and open problems

Entropy methods provide a framework which is extremely convenient for solving
some fundamental variational problems and identifying the optimal constant in
the corresponding functional inequalities. In some cases, evolving the
functional by a nonlinear diffusion amounts to test the equation by a
complicated expression involving a second order differential operator, powers
of the solution and eventually homogeneous weights, which gives rises to
delicate and complex computations. The use of a flow and of the corresponding
entropy functional provides a framework to reinterpret the various terms and
order the computations, as in Section 3.2. Identifying the optimality cases
becomes also much easier and the stationarity with respect to the flow even
provides uniqueness results which would be otherwise difficult to guess or to
reduce to more standard techniques. Parabolic equations come with some extra
properties, like regularization effects, which are crucial to obtain the
stability results of Section 2.5. However, its major advantage is to relate
the nonlinear regime with the linearized one around self-similar or stationary
solutions, using quantities which are meaningful in the two cases: from the
gradient flow point of view in the nonlinear regime and in terms of
orthogonality in the linear regime. It is a remarkable feature that the use of
a parabolic flow allows to bypass symmetrization techniques. This opens new
directions of research in non-real valued problems (complex valued functions
in quantum mechanics, in presence of magnetic fields, see for instance [23,
24], or systems of equations).

Entropy methods have been introduced in partial differential equations, see
[5], in order to handle systems of charged particles, with the aim of
obtaining rates in kinetic and related equations. There is a whole area of
research which has emerged during the last 20 years under the name of
_hypocoercivity_ , with $\mathrm{H}^{1}$ methods (see [94]) or
$\mathrm{L}^{2}$ methods (see [58, 4]). In $\mathrm{L}^{2}$ hypocoercive
approaches, there is a simple strategy: if the kinetic equation admits a
diffusion limit (under the appropriate parabolic scaling) whose asymptotic
behaviour is governed by a functional inequality, then the corresponding rates
of convergence or decay can be reimported in the kinetic equation: see for
instance [25, 27, 26]. This is even true for systems with a non-local Poisson
coupling, as shown in [1]. However, identifying sharp rates and relating
nonlinear models with their linearized counterparts, as can be done in the
above examples of linear and nonlinear parabolic equations, is not done yet,
even in the simplest benchmark cases considered in [4].

Let us conclude this paper by some open problems:
$\rhd$ _Unconfined or weakly confined diffusions._ Diffusion equations without
any external potential or with an external potential which is insufficient to
balance the diffusion are now rather well understood and even the hypocoercive
theory in corresponding kinetic equations is essentially under control, see
[26, 27, 25]. In [26, Section 6], improved decay rates are obtained by Fourier
estimates when moments of low order are set to zero. A spectral interpretation
of these moment conditions, in absence of a bounded stationary solution and
the corresponding functional inequalities, written as entropy-entropy
production inequalities, is however still missing. Reinterpreting these
inequalities in terms of spectral gaps would also be very useful. In kinetic
theory, recent results of [2, 32] also point in this direction, with an
explicit role played by a Poincaré-Lions type inequality.
$\rhd$ _Branches of solutions, optimal constants and stability for Sobolev’s
inequality: a conjecture on the sphere._ Let us consider the inequality

$\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\geq\frac{\mu(\lambda)}{p-2}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}\quad\forall\,u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\mu)$
(4.1)

with optimal constant $\mu(\lambda)$. For an optimal function $u$, the
corresponding Euler-Lagrange equation is

$-\,(p-2)\,\Delta\,u+\lambda\,u=u^{p-1}$ (4.2)

under the normalization condition
$\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{p-2}=\mu(\lambda)$. Since the
optimal constant $\mu(\lambda)$ is defined as an infimum (in $u$) of an affine
function of $\lambda$, $\lambda\mapsto\mu(\lambda)$ is a concave function. The
key estimate is the _rigidity_ result: $u\equiv\lambda^{1/(p-2)}$ is the
unique solution of (4.2) if $\lambda\leq d$, and as a consequence we know that
$\mu(\lambda)=\lambda$.

The first positive eigenvalue of the Laplace-Beltrami operator $\Delta$ on
${\mathbb{S}}^{d}$ is $\lambda_{1}=d$. Let $\varphi_{1}$ be an associated
eigenfunction. By taking as a test function
$u_{\varepsilon}(z)=1+\varepsilon\,\varphi_{1}(z)$ in the limit as
$\varepsilon\to 0_{+}$, it is easy to see that $\lambda=\lambda_{1}$ is a
_bifurcation_ point: there is a branch of non-constant solutions emerging from
constant solutions at $\lambda=d$. Hence $d$ can be characterized as the
minimum of $\\{\lambda>0\,:\,\mu(\lambda)<\lambda\\}$, and _symmetry breaking_
, in the sense of non-constant optimal functions, occurs if and only if
$\lambda>d$. See Fig. 2.

For $p<2^{*}$, it is not very difficult to establish the existence of optimal
functions with _antipodal symmetry_ , that is, of solutions which satisfy the
condition

$u(x)=(-x)\quad\forall\,x\in{\mathbb{S}}^{d}\,.$

$\lambda$$\mu$$\mu=\mu(\lambda)$$\mu=\mu_{a}(\lambda)$ Figure 2.  Case of the
sphere in dimension $d=5$: in the subcritical case $p<2^{*}$, the optimal
constant of (4.1) is given by $\lambda\mapsto\mu(\lambda)$. The dashed line is
the branch of constant solutions $u\equiv 1$. The first bifurcation point is
at $\lambda_{1}=d$. From the second bifurcation point, at
$\lambda_{2}=2\,(d+1)$, emerges another branch with a turning point: they
intersect at $\lambda=\kappa_{p}$. The curve $\lambda\mapsto\mu_{a}(\lambda)$
is obtained as the minimum of this branch and of the line $\mu=\lambda$.

Let us denote by $\mathrm{H}^{1}_{a}({\mathbb{S}}^{d},d\mu)$ the subset of
$\mathrm{H}^{1}({\mathbb{S}}^{d},d\mu)$ with such a symmetry. We can
numerically compute the lowest value for which $\lambda=\mu_{a}(\lambda)$
among the antipodal solutions $u_{\lambda}$ of (4.2), where

$\mu_{a}(\lambda)=\frac{(p-2)\,\|{\nabla
u_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u_{\lambda}}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}-\|{u_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}\,.$

This can be done using partial symmetries and a convenient reparametrization
of the branch emerging from $\lambda_{2}$. The branch is also shown on Fig. 2.
The intersection with the line $\mu=\lambda$ is expected to determine the
optimal constant $\kappa_{p}$ in the inequality

$\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\geq\frac{\kappa_{p}}{p-2}\left(\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}-\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)\quad\forall\,u\in\mathrm{H}_{a}^{1}({\mathbb{S}}^{d},d\mu)$
(4.3)

The curve $p\mapsto\kappa_{p}$ is shown on Fig. 3. The loss of compactness as
$p\to 2^{*}$ corresponds to a concentration on two antipodal points. Based on
these numerical computations, we expect that
$\kappa_{2^{*}}=2^{1-2/2^{*}}d=2^{2/d}\,d$. Because of the stereographic
projection and after taking into account the normalization of the volume of
the sphere ($d\mu$ is a probability measure), this supports the conjecture:

_The optimal constant $\kappa$ in (4.3) for $p=2^{*}$, on ${\mathbb{S}}^{d}$,
is $\kappa_{2^{*}}=2^{2/d}\,d$._

Such a conjecture is an analogue of the Bianchi-Egnell estimate (1.2) written
on ${\mathbb{S}}^{d}$ instead of ${\mathbb{R}}^{d}$, where $\mathsf{S}_{d}$ is
the optimal constant in (1.1). It is compatible with the behaviour of the
branches originating from $\lambda_{2}$, whose lower part converges to an
horizontal line as $p\to 2^{*}$. Such a phenomenon is similar to what has been
observed in [47, 46] for Caffarelli-Kohn-Nirenberg inequalities.

$p$$\kappa_{p}$ Figure 3.  A numerical computation of the optimal constant
$\kappa_{p}$ for (4.3) when $d=5$. Only a few points have been computed on the
dashed curve. The value of $\kappa_{1}$ (at $p=1$) is known and equal to
$\lambda_{2}=2\,(d+1)=12$. At $p=2^{*}$, we get that $\kappa_{10/3}\approx
6.59754$.

### Acknowledgments

This work was supported by the Project EFI (ANR-17-CE40-0030) of the French
National Research Agency (ANR). _© 2021 by the author. This paper may be
reproduced, in its entirety, for non-commercial purposes._

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