Abstract
In this paper, we prove weighted quantitative isoperimetric inequalities for the set \(E_{\alpha}= \{(x,y)\in {R}^{h+1}: \vert y \vert <\int_{\arcsin \vert x \vert }^{\frac{\pi}{2}}\sin^{\alpha +1}(t)\,dt, \vert x \vert <1 \}\) in half-cylinders in the Grushin space \({R}^{h+1}\) with density \(\vert x \vert ^{p}\), \(p\geq0\).
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1 Introduction
The study of isoperimetric problems in Carnot-Carathéodory spaces has been an active field over the past few decades. Pansu [1] first proved an isoperimetric inequality of the type \(P_{H}(E)\geq C \vert E \vert ^{\frac{3}{4}}\) (\(C>0\)) in the Heisenberg group \(\mathbb{H}^{1}\) where \(P_{H}(E)\) and \(\vert E \vert \) denote Heisenberg perimeter and Lebesgue volume of E, respectively. In 1983 Pansu [2] conjectured that, up to a null set, a left translation and a dilation, the isoperimetric set in the Heisenberg group \(\mathbb{H}^{1}\) is a bubble set as follows:
The formula defining \(E_{\mathrm{isop}}\) in (1) makes sense in \(\mathbb{H}^{n}\) for \(n\geq2\) and Pansu’s conjecture can be naturally extended to any dimension. Until today Pansu’s conjecture has not completely been solved. It has been only supported by many partial results, where further hypotheses involving regularity or symmetry of the admissible sets are assumed; see [3–8]. For Carnot groups, one can only get an isoperimetric inequality [9] though we know the fact that isoperimetric sets exist [10].
Monti and Morbidelli [11] completely solved the isoperimetric problem in the Grushin plane \({R}^{2}\). Franceschi and Monti [12] studied isoperimetric problem for a class of x-spherical symmetry sets (here if \(h=1\), the assumption of x-spherical symmetry can be removed) in Grushin spaces \({R}^{h+k}\). In particular, they pointed out that when \(k=1\), up to a null set, a vertical translation and a dilation, the x-spherical symmetric isoperimetric set is
In the case of \(\alpha=1\), the set \(E_{\alpha}\) (2) is just the Pansu sphere in the Heisenberg group.
On the other hand, manifolds with density, a new category in geometry, have been widely studied. They arise naturally in geometry as quotients of Riemannian manifolds, in physics as spaces with different media, in probability as the famous Gauss space and in a number of other places as well (see [13, 14]). Morgan and Pratelli [15] studied the isoperimetric problems in Euclidean spaces \({R}^{n}\) with density; see [16–20] and the references therein. The weighted Sobolev and Poincaré inequalities for Hörmander’s vector fields were well studied in [21–23]. The weighted isoperimetric-type and Sobolev-type inequalities for hypersurfaces in the Carnot group with density were obtained in [24]. In [25] He and Zhao proved that the set \(E_{\alpha}\) is also a weighted x-spherical symmetry isoperimetric set in the Grushin space \({R}^{h+1}\) with density \(\vert x \vert ^{p}\), \(p> -h+1\).
Very recently, Franceschi et al. [26] obtained quantitative isoperimetric inequalities for the bubble set \(E_{\mathrm{isop}}\) in half-cylinders in Heisenberg groups by the construction of sub-calibrations.
Motivated by the nice work mentioned above, in this paper we consider the quantitative isoperimetric inequalities for the set \(E_{\alpha}\) in half-cylinders in the Grushin space \({R}^{h+1}\) with density \(\vert x \vert ^{p}\), \(p\geq0\). These inequalities show that the weighted volume distance of a set F from the set \(E_{\alpha}\) with the same weighted volume is controlled in terms of the difference of the weighted α-perimeter of F and the weighted α-perimeter of \(E_{\alpha}\). We get the following theorem.
Theorem 1.1
Let F be any measurable set in the Grushin space \({R}^{h+1}\) with density \(e^{\phi}= \vert x \vert ^{p}\), \(p\geq0\), where F satisfies \(V_{\phi}(F)=V_{\phi}(E_{\alpha})\). Let \(C_{\varepsilon}= \{(x,y)\in {R}^{h+1}: \vert x \vert <1, y>y_{\varepsilon}\}\) be half-cylinders, where \(0\leq\varepsilon<1\) and \(y_{\varepsilon}=f(1-\varepsilon)\) with \(f(r)=\int_{\arcsin r}^{\frac{\pi}{2}}\sin^{\alpha+1}(t)\,dt\).
-
(i)
If \(F\triangle E_{\alpha}\subset\subset C_{0}\), then we have
$$P_{\alpha,\phi}(F)-P_{\alpha,\phi}(E_{\alpha})\geq\frac{h+p}{120\omega _{h}^{2}}V_{\phi}(E_{\alpha}\triangle F)^{3}. $$ -
(ii)
If \(F\triangle E_{\alpha}\subset\subset C_{\varepsilon}\) with \(0<\varepsilon<1\), then we have
$$P_{\alpha,\phi}(F)-P_{\alpha,\phi}(E_{\alpha})\geq\frac{\sqrt{\varepsilon }(h+p)}{8 [(1-\varepsilon)^{\alpha+h}+(1-\varepsilon)^{h}\sqrt {\varepsilon} ]\omega_{h}}V_{\phi}(E_{\alpha}\triangle F)^{2}. $$Here \(P_{\alpha,\phi}(E)=\sup \{\int_{E} \operatorname{div}_{\alpha}( \vert x \vert ^{p}\varphi)\,dx\,dy:\varphi\in C_{c}^{1} ({R}^{h+1};{R}^{h+1}),\max \vert \varphi \vert \leq1 \}\) and \(V_{\phi}(E)=\int_{E} \vert x \vert ^{p} \,dx\,dy\) are called the weighted α-perimeter and the weighted volume of E, respectively. Finally \(\omega_{h}\) denotes the Euclidean volume of the unit ball.
When \(p=0\) in Theorem 1.1, we can obtain the quantitative isoperimetric inequalities for the set \(E_{\alpha}\) in half-cylinders in Grushin spaces.
2 Preliminaries
The Grushin space \({R}^{h+1}=\{(x,y):x\in{R}^{h}, y\in{R}\}\) is a Carnot-Carathéodory space with a system of vector fields
where \(\alpha>0\) is a given real number and \(\vert x \vert \) is the standard Euclidean norm of x.
The α-perimeter of a measurable set \(E\subset {R}^{h+1}\) in an open set \(A\subset{R}^{h+1}\) is defined as
where the α-divergence of the vector field \(\varphi:A\rightarrow{R}^{h+1} \) is given by
If \(P_{\alpha}(E;A)<\infty\), by the Riesz representation theorem there exist a positive Radon measure \(\mu_{E}\) on A and a \(\mu_{E}\)-measurable function \(v_{E}:A\rightarrow{R}^{h+1}\) such that \(\vert v_{E} \vert =1\) \(\mu_{E}\)-a.e. on A and the generalized Gauss-Green formula
holds for all \(\varphi\in C_{c}^{1}(A; {R}^{h+1})\). Here and hereafter, \(\langle\cdot,\cdot\rangle\) denotes the standard Euclidean scalar product. The measure \(\mu_{E}\) is called α-perimeter measure and the function \(v_{E}\) is called measure theoretic inner unit α-normal of E.
Now we endow the Grushin space \({R}^{h+1}\) with density \(e^{\phi}\) and define the weighted α-perimeter of a measurable set \(E\subset{R}^{h+1}\) in an open set \(A\subset{R}^{h+1}\) as
where \(\operatorname{div}_{\alpha,\phi}\varphi=e^{-\phi}\operatorname {div}_{\alpha}(e^{\phi}\varphi) \) is called the weighted α-divergence of φ.
By the definition of \(\operatorname{div}_{\alpha,\phi }\varphi\), (4) can also be rewritten as
If \(P_{\alpha,\phi}(E;A)<\infty\), then by (3) we have
where \(dV_{\phi}=e^{\phi}\,dx\,dy\) is the weighted volume measure and \(d\mu_{E,\phi}=e^{\phi}\,d\mu_{E}\) is called the weighted α-perimeter measure. For any open set \(A\subset{R}^{h+1}\), we have \(P_{\alpha,\phi}(E;A)=\mu_{E,\phi}(A)\). When \(A={R}^{h+1}\), let \(P_{\alpha,\phi}(E)=P_{\alpha,\phi}(E;{R}^{h+1})\).
Let Σ be a hypersurface in the Grushin space \({R}^{h+1}\) with density \(e^{\phi}\). Σ can be locally given by the zero set of a function \(u\in C^{1}\) such that \(\vert \nabla_{\alpha}u \vert \neq0\) on Σ, where \(\nabla_{\alpha}u=(X_{1}u,\ldots,X_{h}u,Yu)\) is called the α-gradient of u. For a set \(E= \{(x,y)\in{R}^{h+1}:u(x,y)>0 \}\), the inner unit α-normal in equation (6) is given on \(\Sigma=\partial E\) by the vector
Then we define the weighted α-mean curvature of Σ as
Remark 2.1
Noticing that the α-mean curvature of Σ is defined by \(H_{\Sigma}=-\frac{1}{h}\operatorname{div}_{\alpha}v_{E}\), then from (7) we have
To prove Theorem 1.1, we need the following lemma.
Lemma 2.1
Let the Grushin space \({R}^{h+1}\) be endowed with density \(e^{\phi}= \vert x \vert ^{p}\). For any \(0\leq\varepsilon<1\), let \(C_{\varepsilon}= \{(x,y)\in{R}^{h+1}: \vert x \vert <1, y>y_{\varepsilon}\}\) be half-cylinders, where \(y_{\varepsilon}=f(1-\varepsilon)\) with \(f(r)=\int_{\arcsin r}^{\frac{\pi}{2}}\sin^{\alpha+1}(t)\,dt\). There exists a continuous function \(u:C_{\varepsilon}\rightarrow{R}\) with level sets \(\Sigma_{\varepsilon}= \{(x,y)\in C_{\varepsilon}:u(x,y)=s \}\), \(s\in{R}\), such that
-
(i)
\(u\in C^{1}(C_{\varepsilon}\cap E_{\alpha})\cap C^{1}(C_{\varepsilon}\setminus E_{\alpha})\) and \(\frac{\nabla_{\alpha}u(x,y)}{ \vert \nabla_{\alpha}u(x,y) \vert }\) is continuously defined on \(C_{\varepsilon}\setminus\{x=0\}\);
-
(ii)
\(\bigcup_{s>1}\Sigma_{s}=C_{\varepsilon}\cap E_{\alpha}\) and \(\bigcup_{s\leq 1}\Sigma_{s}=C_{\varepsilon}\setminus E_{\alpha}\);
-
(iii)
\(\Sigma_{s}\) is a hypersurface of class \(C^{2}\) with constant weighted α-mean curvature, that is,
$$H_{\Sigma_{s},\phi}=\frac{1}{s}\biggl(1+\frac{p}{h}\biggr)\quad\textit{for } s>1 $$and
$$H_{\Sigma_{s},\phi}=1+\frac{p}{h}\quad\textit{for } s\leq1; $$ -
(iv)
for any point \((x,f( \vert x \vert )-y )\in\Sigma_{s}\) with \(s>1\), we have
$$ 1-\frac{h}{h+p}H_{\Sigma_{s},\phi} \bigl(x,f\bigl( \vert x \vert \bigr)-y \bigr)\geq\frac{1}{5}y^{2}\quad\textit{when } \varepsilon=0 $$(8)and
$$ 1-\frac{h}{h+p}H_{\Sigma_{s},\phi} \bigl(x,f\bigl( \vert x \vert \bigr)-y \bigr)\geq\frac{\sqrt {\varepsilon}}{(1-\varepsilon)^{\alpha}+\sqrt{\varepsilon}}y \quad\textit{when } 0< \varepsilon< 1. $$(9)
Proof
The profile function of the set \(E_{\alpha}\) is the function \(f:[0,1]\rightarrow{R}\),
Its first and second derivatives are
We define the function \(g:[0,1]\rightarrow{R}\),
Its derivative is
Now we construct a foliation of \(C_{\varepsilon}\). In \(C_{\varepsilon}\setminus E_{\alpha}\), the leaves \(\Sigma_{s}\) of the foliation are vertical translations of the top part of the boundary \(\partial E_{\alpha}\). In \(C_{\varepsilon}\cap E_{\alpha}\), the leaves \(\Sigma_{s}\) are constructed as follows: the surface \(\partial E_{\alpha}\) is dilated by a factor larger than 1 where dilation is defined by \((x,y)\rightarrow(\lambda x,\lambda^{\alpha+1}y)\) \((\forall\lambda>0)\), and then it is translated downwards in such a way that the surface \(\{y=y_{\varepsilon}=f(1-\varepsilon)\}\) is also the leaf at last.
We construct a function u on the set \(C_{\varepsilon}\setminus E_{\alpha}\) as
Let \(\Sigma_{s}= \{(x,y)\in C_{\varepsilon}\setminus E_{\alpha}:u(x,y)=s \} \). Then we have \(s\leq1\) and \(\Sigma_{1}=\partial E_{\alpha}\). From (14), we know \(u\in C^{1}(C_{\varepsilon}\setminus E_{\alpha})\) and \(\bigcup_{s\leq 1}\Sigma_{s}=C_{\varepsilon}\setminus E_{\alpha}\).
In the following we will define the function u on the set \(D_{\varepsilon}=C_{\varepsilon}\cap E_{\alpha}\). Setting \(r= \vert x \vert \) and \(r_{\varepsilon}=1-\varepsilon\), we let \(F_{\varepsilon}:D_{\varepsilon}\times(1,\infty)\rightarrow{R}\) be a function
For any \((x,y)\in D_{\varepsilon}\) we have
On the other hand, using (12) and (13) we have
So there exists a unique \(s>1\) such that \(F_{\varepsilon}(x,y,s)=0\) for any \((x,y)\in D_{\varepsilon}\). Furthermore we can define a function \(u:D_{\varepsilon}\rightarrow{R},s=u(x,y)\) determined by the equation \(F_{\varepsilon}(x,y,s)=0\). Obviously we have \(u\in C^{1}(C_{\varepsilon}\cap E_{\alpha})\) and \(C_{\varepsilon}\cap E_{\alpha}=\bigcup_{s>1}\Sigma_{s}\), where \(\Sigma_{s}= \{(x,y)\in C_{\varepsilon}\cap E_{\alpha}:s=u(x,y) \mbox{ is determined by }F_{\varepsilon}(x,y,s)=0 \}\).
By (15), we find
Using (11), (16) and (17), we obtain
Then we have
and the square length of the α-gradient of u on \(D_{\varepsilon}\) is
Note that \(\vert \nabla_{\alpha}u \vert =0\) if and only if \(x=0\). So for any \((x,y)\in D_{\varepsilon}\) with \(x\neq0\), we have
If \((x,y)\in D_{\varepsilon}\) tends to \((\overline{x},\overline{y})\in \partial E_{\alpha}\) with \(\overline{x}\neq0\) and \(\overline{y}>0\), then \(s=u(x,y)\) converges to 1. From (19), we have
where the right hand side is computed by the definition (14) of u. The above equality shows that \(\frac{\nabla_{\alpha}u}{ \vert \nabla_{\alpha}u \vert }\) is continuous on \(C_{\varepsilon}\setminus\{x=0\}\).
In the case of \(e^{\phi}= \vert x \vert ^{p}\), we get \(\phi=p\ln \vert x \vert \) and \(\nabla_{\alpha}\phi= (\frac{p}{ \vert x \vert ^{2}}x_{1},\ldots,\frac {p}{ \vert x \vert ^{2}}x_{h},0 )\) for \(x\neq0\). From (14), we know that the inner unit α-normal of \(\Sigma_{s}\) with \(s\leq1\) is
So the weighted α-mean curvature \(H_{\Sigma_{s},\phi}\) of \(\Sigma_{s}\) with \(s\leq1\) is given by
From (19) we know that the inner unit α-normal of \(\Sigma_{s}\) with \(s>1\) is
So the weighted α-mean curvature \(H_{\Sigma_{s},\phi}\) of \(\Sigma_{s}\) with \(s>1\) is given by
Fixing a point x with \(\vert x \vert <1-\varepsilon\) and for \(0\leq y< f( \vert x \vert )-y_{\varepsilon}\), we define the function
where \(s\geq1\) is uniquely determined by \((x,f( \vert x \vert )-y)\in \Sigma_{s}\). Then the function \(y\rightarrow h_{x}(y)\) is increasing and \(h_{x}(0)=1\).
for all \(0\leq y< f( \vert x \vert )-y_{\varepsilon}\).
By (13), g is strictly increasing. So \(h_{x}(y)\) satisfies
On the other hand, for any \(s>1\) we have
When \(\varepsilon=0\), we have \(r_{\varepsilon}=1\). So (22) turns into
By (21), we get
Integrating (23) with \(h_{x}(0)=1\), we get
Thus we obtain
Noticing
we have
When \(0<\varepsilon<1\), (22) turns into
So by (21), we have
Integrating (25) with \(h_{x}(0)=1\), we have
Noticing \(y< f( \vert x \vert )-y_{\varepsilon}\leq1\), so we have
□
3 Proof of Theorem 1.1
Let \(u:C_{\varepsilon}\rightarrow{R}\) be the function given by Lemma 2.1 and let \(\Sigma_{s}= \{(x,y)\in C_{\varepsilon}:u(x,y)=s \}\) be leaves of the foliation, \(s\in{R}\). We define the vector field \(X:C_{\varepsilon}\setminus\{x=0\}\rightarrow{R}^{h+1}\) by
Then X satisfies the following properties:
-
(i)
\(\vert X \vert =1\);
-
(ii)
for \((x,y)\in\partial E_{\alpha}\cap C_{\varepsilon}\), we have \(X(x,y)=-v_{E_{\alpha}}(x,y)\) where \(v_{E_{\alpha}}(x,y)\) is the unit inner α-normal to \(\partial E_{\alpha}\);
-
(iii)
for any point \((x,y)\in\Sigma_{s}\) with \(s\leq1\), we have
$$ \operatorname{div}_{\alpha,\phi} X(x,y)=h+p. $$(27)
For any point \((x,y)\in\Sigma_{s}\) with \(s>1\), we have
Let \(F\subset{R}^{h+1}\) be a set with finite weighted α-perimeter such that \(V_{\phi}(F)=V_{\phi}(E_{\alpha})\) and \(F\triangle E_{\alpha}\subset\subset C_{\varepsilon}\). By Theorem 2.2.2 in [27], without loss of generality we can assume that the boundary ∂F of F is \(C^{\infty}\).
For \(\delta>0\), let \(E_{\alpha}^{\delta}= \{(x,y)\in E_{\alpha}: \vert x \vert >\delta \}\). By (28) and (6), we have
Letting \(\delta\rightarrow0^{+}\) and using the Cauchy-Schwarz inequality, we obtain
By a similar computation, we also have
On the other hand, we have
From (29), (30) and (31), we obtain
It is equivalent to
For any x with \(\vert x \vert <1-\varepsilon\), we define the vertical sections \(E_{\alpha}^{x}= \{y:(x,y)\in E_{\alpha}\}\) and \(F^{x}= \{y:(x,y)\in F \}\). By the Fubini theorem, we have
Letting \(m(x)={\mathcal{L}}^{1}(E_{\alpha}^{x}\setminus F^{x})\), where \(\mathcal{ L}^{1}\) denotes 1-dimensional Lebesgue measure, then we obtain
where \(h_{x}(y)=u (x,f( \vert x \vert )-y )\) is the function introduced in (20).
When \(\varepsilon=0\), by (8) in Lemma 2.1 and the Hölder inequality, (34) turns into
When \(0<\varepsilon<1\), by (9) in Lemma 2.1, and the Hölder inequality, (34) turns into
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11371194; No. 11571172) and the Natural Science Foundation of the Anhui Higher Education Institutions of China (No. KJ2017A324). The authors would like to thank the referees for their valuable comments and suggestions which helped to improve the paper.
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He, G., Zhao, P. Weighted quantitative isoperimetric inequalities in the Grushin space \({R}^{h+1}\) with density \(|x|^{p}\) . J Inequal Appl 2017, 162 (2017). https://doi.org/10.1186/s13660-017-1437-5
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DOI: https://doi.org/10.1186/s13660-017-1437-5