Weighted Korn and Poincaré-Korn Inequalities in the Euclidean Space and Associated Operators

Abstract

We prove functional inequalities on vector fields $u: {{\mathbb {R}}}^d \rightarrow {{\mathbb {R}}}^d$ when ${{\mathbb {R}}}^d$ is equipped with a bounded measure $\hbox {e}^{-\phi } \,\mathrm {d}x$ that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix Du, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part $D^s u$ and, in an improved form of the inequality, an additional term $\nabla \phi \cdot u$. We also consider Poincaré-Korn inequalities for estimating a projection of u by $D^s u$ and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential $\phi$ and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, which compares Du and $D^s u$ on a bounded domain.

Appendices

Appendix A: Extensions, Gometric Observations and Motivation from Kinetic Theory

1.1 On the assumptions and some generalizations

Remark 11

The fact that the Poincaré inequality (H3) implies that $\hbox {e}^{-\phi } \,\mathrm {d}x$ has an average $\left<x\right>$ and a variance $\langle |x|^2 \rangle$ is classical (see, e.g., [31, Corollary 3.2]). Indeed (H3) yields a concentration property of $\hbox {e}^{-\phi (x)} \,\mathrm {d}x$ via a concentration function. A direct application of Fatou’s Lemma allows to extend (H3) to the set $W^{1,\infty }$ of uniformly Lipschitz functions, which includes $x\mapsto x_j$ for all $j\in \left\{ 1,\ldots , d\right\}$. This directly gives $\int _{{{\mathbb {R}}}^d} |x|^2\,\hbox {e}^{-\phi (x)} \,\mathrm {d}x \le d \,C_{\mathrm {\scriptscriptstyle P}}$ and the integrability of x w.r.t. $\hbox {e}^{-\phi (x)} \,\mathrm {d}x$ follows by the Cauchy-Schwarz inequality. By induction, we get under (H3) alone that the set of all polynomial functions ${{\mathbb {R}}}[x]$ is included in $L^2$ and even $H^1$.

Remark 12

As a consequence of Remark 11 and of the strong Poincaré inequality (21), we directly get that for all $i,j \in \left\{ 1,\ldots d\right\}$, $\int _{{{\mathbb {R}}}^d} |x_i\,\partial _j \phi (x)|\,\hbox {e}^{-\phi (x)} \,\mathrm {d}x <\infty$. It is indeed sufficient to apply the strong Poincaré inequality to $x \mapsto x_j$, which is in $H^1$. Note that this gives sense to all quantities of Theorem 1, e.g., $\left\| \nabla \phi \cdot R\right\|$ for any infinitesimal rotation R.

Remark 13

There are many sufficient conditions for the Poincaré inequality of Assumption (H3). When $\phi$ is uniformly convex, it is shown in [3] that $C_{\mathrm P}$ is greater or equal than the convexity constant, hence leading to fully explicit estimates in the two main theorems. If we only assume that $\lim _{|x| \mapsto \infty } |\nabla \phi (x)|=+\infty$, then $\Lambda$ is in fact an operator with compact resolvent, hence with discrete spectrum, and (H3) follows. Another, less stringent, sufficient condition on $\phi$ is $\liminf _{|x| \rightarrow \infty } \big (\frac{1}{2} |\nabla \phi (x)|^2-\Delta \phi (x)\big ) >c$ for some $c>0$ (it implies the Poincaré inequality from the Persson-Agmon formula of [36] or [1, Theorem 3.2]). We note that this last assumption is satisfied by any regular function which coincides with $x \mapsto \alpha \,|x|+\beta$ outside of a large centred ball, where $\alpha$ and $\beta$ are normalization constants.

Remark 14

Assumptions (H1)–(H2)–(H3) may be satisfied in other geometries than the one of the whole Euclidean space ${{\mathbb {R}}}^d$. In particular, given an open, smooth, bounded and connected subset $\Omega$ of ${{\mathbb {R}}}^d$, we observe that these hypotheses are satisfied by the potential $\phi (x)=\exp \big (1/d^2(x,\partial \Omega )\big )$, where $d(x,\partial \Omega )$ denotes the usual Euclidean distance from x to $\partial \Omega$. Here ${{\mathbb {R}}}^d$ is replaced by $\Omega$ equipped with the measure $\hbox {e}^{-\phi (x)} \,\mathrm {d}x$. It is an open question to understand how our results could be extended to usual boundary problems with potentials mimicking walls at the boundary of $\Omega$.

1.2 Rigidity constants and defects of axisymmetry

In the proofs of Theorems 1 and 4 (see also “Appendix B.5”), we used the two rigidity constants $C_{\mathrm {\scriptscriptstyle RV}}$ and $C_{\mathrm {\scriptscriptstyle RV0}}$ defined by (11) and (17), depending on the level of regularity in each case, to measure the defects of axisymmetry (note that $C_{\mathrm {\scriptscriptstyle RV}}\le C_{\mathrm {\scriptscriptstyle RV0}}$ since $\Lambda \ge \mathrm {Id}$). We used also $C_{\mathrm {\scriptscriptstyle RD}}$ defined in (12) but remark that $C_{\mathrm {\scriptscriptstyle RV}}$ and $C_{\mathrm {\scriptscriptstyle RD}}$ are not directly comparable either, because ${\mathfrak {M}}_\phi ^c \ne D {{\mathcal {R}}}_\phi ^c$. Other ways of measuring the default of axisymmetry of the potential $\phi$ can be considered.

$\textcircled {1}$ One can consider, again, a rigidity of vector fields constant, but this time defined alternatively by

$$\begin{aligned} C_{\mathrm {\scriptscriptstyle RVL}}^{-1}=\min _{ A\,x \in {{\mathcal {R}}}_\phi ^c \setminus \left\{ 0\right\} } \frac{\left\| \nabla \phi \cdot A\,x\right\| ^2}{\left\| A\,x\right\| ^2}\quad \text{ when }\quad {{\mathcal {R}}}_\phi ^c \ne \left\{ 0\right\} \quad \text{ and } \quad C_{\mathrm {\scriptscriptstyle RVL}}=0\quad \text{ otherwise }\,. \end{aligned}$$

This leads to the modified Poincaré-Korn inequality

$$\begin{aligned}&\inf _{A\,x\in {{\mathcal {R}}}_\phi }\Vert u- \left<u\right>-A\,x\Vert ^2=\Vert u- \left<u\right>- {{\mathbb {P}}}_\phi (u)\Vert ^2 \nonumber \\&\quad \le C_{\mathrm {\scriptscriptstyle PKL}}'\,\Vert D^s u\Vert ^2 +2\,C_{\mathrm {\scriptscriptstyle RVL}}\,\Vert \nabla \phi \cdot (u-\left<u\right>)\Vert ^2 \end{aligned}$$

(48)

with an explicit bound for the constant $C_{\mathrm {\scriptscriptstyle PKL}}'$ using (9) and the method of proof of (13). Once more, the existence of $C_{\mathrm {\scriptscriptstyle RVL}}$ follows from the injectivity of $A\,x \mapsto \nabla \phi \cdot A\,x$ on ${{\mathcal {R}}}_\phi ^c$ and the fact that ${{\mathcal {R}}}_\phi ^c$ is of finite dimension. The main advantage of this approach is to preserve a continuity property with respect to axisymmetry, which can be stated as follows: a small perturbation of a radial potential $\phi$ gives rise to a small constant $C_{\mathrm {\scriptscriptstyle RVL}}$, the limiting case being ${{\mathcal {R}}}_\phi ^c=\left\{ 0\right\}$ and $C_{\mathrm {\scriptscriptstyle RVL}}=0$. The main drawback is that the symmetric operator associated to (48) is neither local nor differential because of the term $\left<u\right>$ which appears in the right-hand side of (48).

$\textcircled {2}$ In a bounded domain $\Omega \subset {{\mathbb {R}}}^d$, with flat metric (i.e., for a constant potential $\phi$) considered in [12], the authors use Grad’s number. Let us explain how to adapt this method in our context under, e.g., the additional condition

$$\begin{aligned} \lim _{|x| \rightarrow \infty }\frac{D^2\phi (x)}{\lfloor \nabla \phi (x)\rceil ^2}=0\,. \end{aligned}$$

This property implies that the multiplication operator by $D^2 \phi$ is relatively compact with respect to $-\Delta _\phi$ acting on vector fields, with essential spectrum in $[C_{\mathrm {\scriptscriptstyle P}}, \infty )$. The spectrum in $[0,C_{\mathrm {\scriptscriptstyle P}})$ is then a pure point spectrum and the kernel is finite dimensional. For any antisymmetric matrix $A$, there exists an affine space $\mathfrak {V}_A$ of functions $v\in H^1$ solving the Witten-Hodge problem

$$\begin{aligned} {{\,\mathrm{div_{\phi }}\,}}v=0\,,\quad D^av=A\,. \end{aligned}$$

The Witten-Hodge inequality asserts that

$$\begin{aligned} \inf _{v \in \mathfrak {V}_A}\Vert D^s v\Vert ^2 \le c_{\mathrm H}\,|A|^2 \end{aligned}$$

(49)

for some constant $c_{\mathrm H}\in (0,\infty )$. The reverse inequality amounts to the existence of Grad’s number such that

$$\begin{aligned} C_{\mathrm G}^{-1} :=\inf _{A\in {\mathfrak {M}}_\phi ^c,\,|A|=1,\,v \in \mathfrak {V}_A}\;\Vert D^sv\Vert ^2. \end{aligned}$$

The existence of $C_{\mathrm G}$ as well as a quantitative positive lower bound could be establish using mass transport theory exactly as in [12]. Of course, $C_{\mathrm G}$ is well defined only when ${{\mathcal {R}}}_\phi ^c\ne \{0\}$, i.e., under the condition that $\phi$ is not radially symmetric. Inequality (49) is natural in differential geometry and more specifically in De Rham cohomology theory: we refer to [2, 21] for further developments on this topic. In bounded domains, how to measure the symmetry defect by Grad’s number in view of Korn type inequalities is at the core of [12] but has also been studied in [18]. This approach differs from ours. Using $C_{\mathrm G}$ and (8), the inequality (13) can be proved along a similar strategy as in [12], although with different constants.

1.3 An elementary application in kinetic theory

The main motivation for this paper comes from kinetic equations involving a confining potential studied in [7]. Also see [12, Section 2] and [4, 15] for applications of Korn inequalities to kinetic equations. As an example, let us consider the linear relaxation model of BGK-type

$$\begin{aligned} \partial _tf+v\cdot \nabla _x f -\nabla _x \phi \cdot \nabla _v f =\mathcal Lf :=G_f-f, \end{aligned}$$

(50)

where f(t, x, v) is an unknown distribution function for a system of particles depending on time $t \ge 0$, position $x\in {{\mathbb {R}}}^d$ and velocity $v\in {{\mathbb {R}}}^d$, and where $G_f$ is defined by

$$\begin{aligned} G_f:=(\rho +\mathbf {u}\cdot v)\,\mu \quad&\text { where }\quad \rho (t,x) :=\int _{{{\mathbb {R}}}^d} f(t,x,v)\,\,\mathrm {d}v\,, \\&\text { and }\quad \mathbf {u}(t,x) :=\int _{{{\mathbb {R}}}^d}v\,f(t,x,v)\,\,\mathrm {d}v\,. \end{aligned}$$

Here $\mu (v):=(2\,\pi )^{-d/2}\,\hbox {e}^{-|v|^2/2}$ while $\rho$ and $\mathbf{u}$ are respectively the macroscopic density and the average velocity associated with f. The collision kernel admits $d+1$ conserved moments, in the sense that $\int _{{{\mathbb {R}}}^d}\mathcal Lf(t,x,v)\,\,\mathrm {d}v=0=\int _{{{\mathbb {R}}}^d}v_i\,\mathcal Lf(t,x,v)\,\,\mathrm {d}v$ for any $i=1,\ldots ,d$ and $f \in L^1((1+|v|)\mathrm{d} v)$.

A natural question is to look for equilibria of (50). A quick glance at the equation shows that ${{\mathcal {M}}}(x,v) :=\hbox {e}^{-\phi (x)}\,\mu (v)$ is one of them. Korn inequalities provide us with a complete answer.

Proposition 15

Under Assumptions (H1)–(H2)–(H3), all equilibria of (50) in $L^2({\mathcal {M}}^{-1}\,\mathrm {d}x\,\mathrm {d}v)$ take the form $f(x,v)=\big (( R(x)\cdot v)+c\big )\,{{\mathcal {M}}}$ for some $R \in {{\mathcal {R}}}_\phi$ and $c \in {{\mathbb {R}}}$.

Proof

Write $f=h\,{\mathcal {M}}$ with $h \in L^2({\mathcal {M}}\,\mathrm {d}x\,\mathrm {d}v)$, $\rho =r\,\hbox {e}^{-\phi }$, $\mathbf{u}=u\,\hbox {e}^{-\phi }$, so that equation (50) reads

$$\begin{aligned} \partial _th+v\cdot \nabla _x h -\nabla _x \phi \cdot \nabla _v h = L(h) :=h-r-u\cdot v. \end{aligned}$$

(51)

The restriction of L to $L^2(\mu \,\mathrm {d}v)$ is $L=-\,\Pi ^\bot$ where $\Pi$ is the orthogonal projection onto $\mathrm {Span} \{1, v_1, \ldots , v_d\}$. We compute

$$\begin{aligned}&\frac{d}{dt}\iint _{{{\mathbb {R}}}^d\times {{\mathbb {R}}}^d}|h|^2\,{{\mathcal {M}}}\,\mathrm {d}x\,\mathrm {d}v = 2\iint _{{{\mathbb {R}}}^d\times {{\mathbb {R}}}^d}(L h)\,h\,{{\mathcal {M}}}\,\mathrm {d}x\,\mathrm {d}v \\&\quad = -\,2 \iint _{{{\mathbb {R}}}^d \times {{\mathbb {R}}}^d} \left| \Pi ^\bot h \right| ^2\, {{\mathcal {M}}}\,\mathrm {d}x \,\mathrm {d}v \end{aligned}$$

and deduce that any stationary solution of (51) takes the form $h(x,v)=r(x)+u(x)\cdot v$. Equation (51) then reads

$$\begin{aligned} v\cdot \nabla _x(r+u\cdot v)=\nabla _x\phi \cdot u. \end{aligned}$$

Integrating the latter equation against respectively 1, $v_i$ and $v_i\,v_j$ with $i \not =j$ in $L^2(\mu \,\mathrm {d}v)$ yields

$$\begin{aligned} \mathrm{i)}\quad \nabla _x\cdot u-\nabla _x \phi \cdot u=0,\quad \mathrm{ii)}\quad D^s u=0,\quad \mathrm{iii)}\quad \nabla r=0. \end{aligned}$$

From iii) we get that there exists $c \in {{\mathbb {R}}}$ such that $r=c$. As for i), an integration by parts gives

$$\begin{aligned}&0=\int _{{{\mathbb {R}}}^d} (\nabla _x\cdot u-\nabla _x \phi \cdot u) \left<u\right>\cdot x\,\hbox {e}^{-\phi (x)} \,\mathrm {d}x= -\,\int _{{{\mathbb {R}}}^d} u\cdot \nabla \left( \left<u\right>\cdot x\right) \,\hbox {e}^{-\phi (x)} \,\mathrm {d}x\\&\quad = -\,\int _{{{\mathbb {R}}}^d} u\cdot \left<u\right>\,\hbox {e}^{-\phi (x)} \,\mathrm {d}x=-\,\left<u\right>^2, \end{aligned}$$

so that $\left<u\right>=0$. Note also that taking the trace in ii) yields $\nabla _x\cdot u=0$ so that i) reads $\nabla _x \phi \cdot u=0$. Using this and (19) in Theorem 4 shows that $u=R$ with $R={{\mathbb {P}}}_\phi (u)\in {{\mathcal {R}}}_\phi$. Hence $h=R(x)\cdot v+c$ and $f(x,v)=\big (( R(x)\cdot v)+c\big )\,{{\mathcal {M}}}$. The reciprocal is straightforward, which completes the proof. $\square$

Appendix B: Additional Details on Computations

1.1 Functions, derivatives and projections

We denote by f a generic scalar function on ${{\mathbb {R}}}^d$ and by $u:{{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d$ a generic vector field, so that $\nabla f=(\partial _if)_{i=1}^d$ is a vector field and $Du=(\partial _ju_i)_{i,j=1}^d$ takes values in ${\mathfrak {M}}$. The symmetric and the antisymmetric differentials of u, respectively $D^su=\big ((D^su)_{ij}\big )_{i,j=1}^d$ and $D^au=\big ((D^au)_{ij}\big )_{i,j=1}^d$ are defined by

$$\begin{aligned} (D^su)_{ij}:=\tfrac{1}{2}\left( \partial _ju_i+\partial _iu_j\right) \quad \text{ and }\quad (D^au)_{ij}:=\tfrac{1}{2}\left( \partial _ju_i-\partial _iu_j\right) \end{aligned}$$

so that $D^su+D^au=Du$.

The orthogonal projection ${{\mathbb {P}}}$ of vector-valued functions is defined as follows. Let $(A_{ij})_{1\le i<j\le d}$ be a basis of $\mathfrak M^a$ whose elements are

$$\begin{aligned} A_{ij}=\big ((\delta _{ij}-\delta _{ji})\,\delta _{ki}\,\delta _{j\ell } \big )_{k,\ell =1}^d \end{aligned}$$

and $(R_{ij})_{1\le i<j\le d}$ the orthonormal (in the $L^2$ sense) basis of ${\mathcal {R}}$ given by

$$\begin{aligned} R_{ij}(x)=Z_{ij}^{-1}\,A_{ij}\,x \end{aligned}$$

whose coordinates are all 0 except the ith and the jth ones, with respective values $-x_j/Z_{ij}$ and $x_i/Z_{ij}$, i.e.,

$$\begin{aligned} R_{ij}(x)^\perp =Z_{ij}^{-1}\,\big (0,\ldots ,0,-x_j,0, \ldots ,0,x_i,0,\ldots ,0\big )\,, \end{aligned}$$

and where the normalization constant is $Z_{ij}=\left( \int _{{{\mathbb {R}}}^d}(x_i^2+x_j^2)\,\hbox {e}^{-\phi (x)}\,\mathrm {d}x\right) ^{1/2}$. With these notations, ${{\mathbb {P}}}u$ is the vector field

$$\begin{aligned} x\mapsto {{\mathbb {P}}}u(x):=\sum _{1\le i<j\le d}{\mathsf {c}}_{ij}\,R_{ij}(x), \end{aligned}$$

where the coefficients are computed, for all integers i, j such that $1\le i<j\le d$, as

$$\begin{aligned} \textstyle {\mathsf {c}}_{ij}= \int _{{{\mathbb {R}}}^d}u(x)\cdot R_{ij}(x)\,\hbox {e}^{-\phi (x)}\,\mathrm {d}x= \frac{1}{Z_{ij}}\int _{{{\mathbb {R}}}^d}\left( x_i\,u_j(x)-x_j\,u_i(x)\right) \, \hbox {e}^{-\phi (x)}\,\mathrm {d}x\,. \end{aligned}$$

The orthogonal projection $\mathfrak {P}$ of a matrix-valued function ${\mathfrak {F}}$ is defined as

$$\begin{aligned} \mathfrak {P}\,{\mathfrak {F}}:=\sum _{1\le i<j\le d}{\mathsf {d}}_{ij}\,A_{ij} \end{aligned}$$

where the coefficients are computed, for all integers i, j such that $1\le i<j\le d$, as

$$\begin{aligned} \textstyle {\mathsf {d}}_{ij}= \frac{1}{2}\int _{{{\mathbb {R}}}^d}\mathfrak F(x):A_{ij}\,\hbox {e}^{-\phi (x)}\,\mathrm {d}x= \frac{1}{2}\int _{{{\mathbb {R}}}^d} \big (\mathfrak F_{ij}(x)-{\mathfrak {F}}_{ji}(x)\big )\, \hbox {e}^{-\phi (x)}\,\mathrm {d}x\,. \end{aligned}$$

As a a consequence, we deduce that $\mathfrak {P}{\mathfrak {F}}=\left<\mathfrak F^a\right>$ and $\mathfrak {P}(Du) = \left<D^a u\right>$ for any $u \in H^1$.

A matrix $A\in D{{\mathcal {R}}}_\phi ^c$ is such that $A\in {\mathfrak {M}}^a$ and for any $B\in D{{\mathcal {R}}}_\phi \subset {\mathfrak {M}}^a$,

$$\begin{aligned} \textstyle 0=\int _{{{\mathbb {R}}}^d}A\,x\cdot B\,x\,\hbox {e}^{-\phi (x)}\,\mathrm {d}x= \sum _{i,j,k=1}^dA_{ij}\,B_{ik} \int _{{{\mathbb {R}}}^d}x_j\,x_k\,\hbox {e}^{-\phi (x)}\,\mathrm {d}x\,. \end{aligned}$$

A matrix $A\in {\mathfrak {M}}_\phi ^c$ is such that $A\in {\mathfrak {M}}^a$ and for any $B\in D{{\mathcal {R}}}_\phi ={\mathfrak {M}}_\phi \subset {\mathfrak {M}}^a$,

$$\begin{aligned} \textstyle 0=\int _{{{\mathbb {R}}}^d}A:B\,\hbox {e}^{-\phi (x)}\,\mathrm {d}x= A:B=\sum _{i,j}^dA_{ij}\,B_{ij}. \end{aligned}$$

Based on these two definitions, it is clear that $D{{\mathcal {R}}}_\phi ^c$ and ${\mathfrak {M}}_\phi ^c$ generically differ.

1.2 Operators

Let us give some details on the differential operators $-\Delta _\phi$ and $-\Delta _S$ associated respectively with the quadratic forms $f\mapsto \Vert \nabla f\Vert ^2$ and $u\mapsto \Vert D^s u\Vert ^2$.

$\rhd$ Using $\nabla _{\!\phi }u := \nabla \cdot u- \nabla \phi \cdot u$, the Witten-Laplace operator $\Delta _\phi$ on functions is such that $\Vert \nabla f\Vert ^2=(f,-\Delta _\phi f)$ and takes the form

$$\begin{aligned} \Delta _\phi f= \hbox {e}^\phi \,\nabla \cdot \left( \nabla f\,\hbox {e}^{-\phi }\right) = \nabla _{\!\phi }\cdot \nabla f = \Delta f - \nabla \phi \cdot \nabla f\,. \end{aligned}$$

$\rhd$ By definition of $D^su$ and using integration by parts, we have

$$\begin{aligned} \begin{array}{rl} (-\Delta _S u,u)&{}{}=2\int _{{{\mathbb {R}}}^d}|D^su|^2\,\text{ e}^{-\phi }\,\mathrm {d}x=\frac{1}{2}\sum _{i,j=1}^d \int _{{{\mathbb {R}}}^d}\left( \partial _iu_j+\partial _ju_i\right) ^2\text{ e}^{-\phi }\,\mathrm {d}x\\ &{}{}=-\frac{1}{2}\sum _{i,j=1}^d\int _{{{\mathbb {R}}}^d}u_j\, \partial _i\left( (\partial _iu_j+\partial _ju_i)\,\text{ e}^{-\phi }\right) \,\mathrm {d}x \\ &{}{}\quad -\frac{1}{2}\sum _{i,j=1}^d\int _{{{\mathbb {R}}}^d}u_i\, \partial _j\left( (\partial _iu_j+\partial _ju_i)\, \text{ e}^{-\phi }\right) \,\mathrm {d}x\\ &{}{}=-\sum _{i,j=1}^d\int _{{{\mathbb {R}}}^d}u_j\, \partial _i\left( (\partial _iu_j+\partial _ju_i)\, \text{ e}^{-\phi }\right) \,\mathrm {d}x\\ &{}{}=-\sum _{i=1}^d\int _{{{\mathbb {R}}}^d}\big (u_i\,\Delta u_i+u_i\,\partial _{ij}u_j\big )\,\text{ e}^{-\phi }\,\mathrm {d}x\\ &{}{}\quad +\sum _{i=1}^d\int _{{{\mathbb {R}}}^d}u_i\, \big ((\nabla \phi \cdot \nabla )\,u_i +2\,(D^su\,\nabla \phi )_i\big )\, \text{ e}^{-\phi }\,\mathrm {d}x\\ &{}{}=-\int _{{{\mathbb {R}}}^d}u\cdot \big (\Delta u+\nabla (\nabla \cdot u)-(\nabla \phi \cdot \nabla )\,u -2\,D^su\,\nabla \phi \big )\,\mathrm {d}x\,. \end{array} \end{aligned}$$

so that $-\Delta _S$ is given, for an arbitrary vector field $u\in {{\mathcal {C}}}_c^\infty ({{\mathbb {R}}}^d;{{\mathbb {R}}}^d)$, by

$$\begin{aligned} -\Delta _S\,u=-\,D^s_\phi \cdot D^su= -\big (\Delta u+\nabla (\nabla \cdot u) -(\nabla \phi \cdot \nabla )\,u-2\,D^su\,\nabla \phi \big )\,. \end{aligned}$$

1.3 Gaussian measure

In the normalized centred Gaussian case $\phi (x)=\frac{1}{2}\,|x|^2+\frac{d}{2}\,\ln (2\pi )$ corresponding to (5), the basic constants are $C_{\mathrm {\scriptscriptstyle P}}=1$ (which is the optimal constant in the Gaussian Poincaré inequality), either $C_\phi =1+4\,d$ and $C_\phi '=4\,\sqrt{d\,(1+4\,d)}$ if $d\ge 2$, or $C_\phi =8$ and $C_\phi '=8\,\sqrt{2}$ if $d=1$, as a limit case.

Let $u(x)=(1-x_2^2,x_1\,x_2,0,\ldots 0)^\perp$. By elementary computations, we find that

$$\begin{aligned}&Du= \begin{pmatrix} \begin{array}{cc} 0&{}-\,2\,x_2\\ x_2&{}x_1\end{array}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{\mathbf{0}} \end{pmatrix},\quad D^su= \begin{pmatrix} \begin{array}{cc}0&{}-\,\frac{1}{2}\,x_2\\ -\,\frac{1}{2}\,x_2&{}x_1\end{array}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{\mathbf{0}} \end{pmatrix},\\&D^au= \begin{pmatrix} \begin{array}{cc}0&{}-\,\frac{3}{2}\,x_2\\ \frac{3}{2}\,x_2&{}0\end{array}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{\mathbf{0}} \end{pmatrix} \end{aligned}$$

where $\mathbf{0}$ denotes $2\times (d-2)$, $(d-2)\times 2$, and $(d-2)\times (d-2)$ null matrices. After integration against the normalized centred Gaussian measure, we have

$$\begin{aligned}&\left<u\right>=0,\quad \Vert u\Vert ^2=3\,,\quad {{\mathbb {P}}}(u)=0\,,\quad \mathfrak {P}(Du)=0=\left<D^au\right>\,,\\&\Vert D^su\Vert ^2=\frac{3}{2}\,,\quad \Vert D^au\Vert ^2=\frac{9}{2}\,, \quad \Vert Du\Vert ^2=6\,. \end{aligned}$$

This proves that $C_{\mathrm {\scriptscriptstyle K}}=4$ in $\Vert Du-\mathfrak {P}(Du)\Vert ^2 \le C_{\mathrm {\scriptscriptstyle K}}\,\Vert D^s u\Vert ^2$ and $C_{\mathrm {\scriptscriptstyle PK}}=2$ in $\Vert u-\left<u\right> - {{\mathbb {P}}}(u)\Vert ^2 \le C_{\mathrm {\scriptscriptstyle PK}}\,\Vert D^s u\Vert ^2$ are both optimal.

1.4 Estimates for the$D(\Lambda )$-toolbox and consequences

Here we give some details on the computation of $C_{\mathrm {\scriptscriptstyle B}}$ in the proof of Proposition 8 in Sect. 3.1. Let $A=\big \Vert \lfloor \nabla \phi \rceil \,\nabla f\big \Vert$, $B=\big \Vert \lfloor \nabla \phi \rceil ^2\,f\big \Vert$, $Z=\Vert \xi \Vert$ and let ${c'}^2=C_\phi '\ge C_\phi =c^2$. Inequalities (36) and (37) amount to

$$\begin{aligned} B\le \tfrac{4}{3}\,c\,A+\tfrac{5}{3}\,{c'}^2\,Z\quad \text{ and } \quad A^2\le B\,Z+\tfrac{1}{2\,c}\,A\,B+\tfrac{{c'}^2}{2\,c}\,A\,Z\,. \end{aligned}$$

Taking the equality case in the first inequality, we find that

$$\begin{aligned} A^2-4\,A\,\big (c+{c'}^2/c\big )\,Z-5\,{c'}^2\,Z^2\le 0 \end{aligned}$$

which means that

$$\begin{aligned} \textstyle A\le \left( 2\,\sigma +\sqrt{4\,\sigma ^2+5}\,\right) c'\,Z \quad \text{ with }\quad \sigma =\frac{c}{c'}+\frac{c'}{c}\ge 2\,. \end{aligned}$$

On $[2,+\infty )$, the function $\sigma \mapsto \sqrt{4\,\sigma ^2+5}/\sigma$ is monotone non-increasing, so that $\sqrt{4\,\sigma ^2+5}\le \frac{1}{2}\,\sqrt{21}\,\sigma$. Using the monotonicity of $c\mapsto c\,\sigma$ and $c\le c'$, we also have $c\,\sigma \le 2\,c'$. As a consequence, we have

$$\begin{aligned}&\textstyle A\le \left( 2+\tfrac{1}{2}\,\sqrt{21}\,\right) c\,\sigma \, \frac{c'}{c}\,Z\le \left( 4+\sqrt{21}\,\right) \frac{{c'}^2}{c}\,Z \le 9\,\frac{{c'}^2}{c}\,Z\,,\\&\textstyle B \le \frac{1}{3}\left( 4\,c\left( 2\,\sigma +\sqrt{4\,\sigma ^2+5}\,\right) +5\,c'\right) c'\,Z\le \left( 7+4\,\sqrt{7/3}\,\right) \le 14\,{c'}^2\,Z\,, \end{aligned}$$

that is, the bounds (38). Moreover, from $\big \Vert D^2f\big \Vert ^2\le \tfrac{C}{d}+\tfrac{1}{2}\left( d+\sqrt{d^2+4\,C}\,\right) \Vert \xi \Vert ^2$ with $C=\tfrac{277}{8}\left( C_\phi -1\right) {C_\phi '}^2$, we deduce that

$$\begin{aligned} C_{\mathrm {\scriptscriptstyle B}}=81\,\tfrac{{C_\phi '}^2}{C_\phi } +196\,{C_\phi '}^2+\tfrac{C}{d}+\tfrac{1}{2}\left( d+\sqrt{d^2+4\,C}\,\right) . \end{aligned}$$

(52)

1.5 Estimates on various constants

The constants $C_\phi$ and $C_\phi '$ appear in (6) as a consequence of (H2) while the Poincaré constant $C_{\mathrm {\scriptscriptstyle P}}$ follows from Assumption (H3). According to (41), the constant in the Poincaré-Lions inequality (22) is given with $C_{\mathrm {\scriptscriptstyle B}}$ as in (52) by $C_{\mathrm {\scriptscriptstyle PL}}=(1+C_{\mathrm {\scriptscriptstyle P}})^2\,\big (1+C_\phi '\,\sqrt{C_{\mathrm {\scriptscriptstyle B}}}/4\big )^2$. From Proposition 5, we know that the strong Poincaré inequality (21) holds for some $C_{\mathrm {\scriptscriptstyle SP}}\le C_\phi \,(1+C_{\mathrm {\scriptscriptstyle P}})$. As for the other constants in Theorems 1 and 2, we learn from the proofs in Sects. 4.1 and 4.2 that

$$\begin{aligned}&C_{\mathrm {\scriptscriptstyle K}}\le 1+4\,C_{\mathrm {\scriptscriptstyle PL}}\,,\quad C_{\mathrm {\scriptscriptstyle PK}}\le C_{\mathrm {\scriptscriptstyle P}}\,C_{\mathrm {\scriptscriptstyle K}}\,, \quad C_{\mathrm {\scriptscriptstyle SPK}}\le C_{\mathrm {\scriptscriptstyle SP}}(C_{\mathrm {\scriptscriptstyle K}}+3\,C_\phi \,C_{\mathrm {\scriptscriptstyle PK}})\,,\\&C_{\mathrm {\scriptscriptstyle PK}}'\le C_{\mathrm {\scriptscriptstyle PK}}+2\,C_{\mathrm {\scriptscriptstyle RV}}\,C_{\mathrm {\scriptscriptstyle SPK}}\quad \text{ and }\quad C_{\mathrm {\scriptscriptstyle K}}'\le C_{\mathrm {\scriptscriptstyle K}}(1+2\,C_{\mathrm {\scriptscriptstyle RD}}\,C_{\mathrm {\scriptscriptstyle SP}})\,. \end{aligned}$$

In Sect. 6.1, Lemma 10, using $C_\phi '':=C_{\varepsilon }$ as in (H2) with ${\varepsilon }=1/(2\,\sqrt{C_{\mathrm {\scriptscriptstyle B}}})$, the constants in (46) are

$$\begin{aligned} C_{\mathrm {\scriptscriptstyle LPL}}\le 4\,(1+C_{\mathrm {\scriptscriptstyle P}})^2\,(1+C_\phi '')^2\quad \text{ and }\quad C_{\mathrm {\scriptscriptstyle RPL}}\le \left( 1+\tfrac{1}{4}\,C_\phi '\,{\textstyle \sqrt{C_{\mathrm {\scriptscriptstyle B}}/C_\phi }}\,\right) ^2\,. \end{aligned}$$

Finally, the constants in Theorem 4 are given by

$$\begin{aligned} C_{\mathrm {\scriptscriptstyle K0}}=1+4\,C_{\mathrm {\scriptscriptstyle LPL}}\,,\quad C_{\mathrm {\scriptscriptstyle PK0}}\le C_{\mathrm {\scriptscriptstyle PL}}\,(1+4\,C_{\mathrm {\scriptscriptstyle LPL}}) \quad \text{ and }\quad C_{\mathrm {\scriptscriptstyle PK0}}' \le C_{\mathrm {\scriptscriptstyle PK0}}(1+2\,C_{\mathrm {\scriptscriptstyle RV0}}\,C_\phi ). \end{aligned}$$