Weighted quantitative isoperimetric inequalities in the Grushin space R h+1 with density x p

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1 He and Zhao Journal of Inequalitie and Application 17 17:16 DOI / R E S E A R C H Open Acce Weighted quantitative ioperimetric inequalitie in the Gruhin pace R h+1 with denity x p Guoqing He 1,* and Peibiao Zhao 1 * Correpondence: hgq11@mail.ahnu.edu.cn 1 School of Science, Nanjing Univerity of Science and Technology, Nanjing, 194, China School of Mathematic and Computer Science, Anhui Normal Univerity, Wuhu, 41, China Abtract In thi paper, we prove weighted quantitative ioperimetric inequalitie for the et E α {x, y R h+1 : y < π arcin x inα+1 t dt, x <1} in half-cylinder in the Gruhin pace R h+1 with denity x p, p. MSC: 5B6; 53C17 Keyword: Gruhin pace; denity; quantitative ioperimetric inequality 1 Introduction The tudy of ioperimetric problem in Carnot-Carathéodory pace ha been an active field over the pat few decade. Panu [1] firt proved an ioperimetric inequality of the type P H E C E 4 3 C >intheheienberggrouph 1 where P H E and E denote Heienberg perimeter and Lebegue volume of E, repectively. In 1983 Panu [] conjectured that, up to a null et, a left tranlation and a dilation, the ioperimetric et in the Heienberg group H 1 i a bubble et a follow: { E iop z, t H 1 : t < 1 } arcco z + z 1 z, z <1. 1 The formula defining E iop in 1 makeeneinh n for n and Panu conjecture can be naturally extended to any dimenion. Until today Panu conjecture ha not completely been olved. It ha been only upported by many partial reult, where further hypothee involving regularity or ymmetry of the admiible et are aumed; ee [3 8].For Carnot group, one can only get an ioperimetric inequality [9] though we know the fact that ioperimetric et exit [1]. Monti and Morbidelli [11] completely olved the ioperimetric problem in the Gruhin plane R. Francechi and Monti [1] tudied ioperimetric problem for a cla of x- phericalymmetryethereifh 1, the aumption of x-pherical ymmetry can be removed in Gruhin pace R h+k. In particular, they pointed out that when k 1,uptoa null et, a vertical tranlation and a dilation, the x-pherical ymmetric ioperimetric et The Author 17. Thi article i ditributed under the term of the Creative Common Attribution 4. International Licene which permit unretricted ue, ditribution, and reproduction in any medium, provided you give appropriate credit to the original author and the ource, provide a link to the Creative Common licene, and indicate if change were made.

2 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page of 1 i { π } E α x, y R h+1 : y < in α+1 t dt, x <1. arcin x In the cae of α 1,theetE α i jut the Panu phere in the Heienberg group. On the other hand, manifold with denity, a new category in geometry, have been widely tudied. They arie naturally in geometry a quotient of Riemannian manifold, in phyic a pace with different media, in probability a the famou Gau pace and in a number of other place a well ee [13, 14]. Morgan and Pratelli [15] tudied the ioperimetric problem in Euclidean pace R n with denity; ee [16 ] and the reference therein. The weighted Sobolev and Poincaré inequalitie for Hörmander vector field were well tudied in [1 3]. The weighted ioperimetric-type and Sobolev-type inequalitie for hyperurface in the Carnot group with denity were obtained in [4]. In [5] HeandZhao proved that the et E α i alo a weighted x-pherical ymmetry ioperimetric et in the Gruhin pace R h+1 with denity x p, p > h +1. Very recently, Francechi et al. [6] obtained quantitative ioperimetric inequalitie for the bubble et E iop in half-cylinder in Heienberg group by the contruction of ubcalibration. Motivated by the nice work mentioned above, in thi paper we conider the quantitative ioperimetric inequalitie for the et E α in half-cylinder in the Gruhin pace R h+1 with denity x p, p. Thee inequalitie how that the weighted volume ditance of a et F from the et E α with the ame weighted volume i controlled in term of the difference of the weighted α-perimeter of F and the weighted α-perimeter of E α. We get the following theorem. Theorem 1.1 Let F be any meaurable et in the Gruhin pace R h+1 with denity e φ x p, p, where F atifie V φ FV φ E α. Let C ε {x, y R h+1 : x <1,y > y ε } be halfcylinder, where ε <1and y ε f 1 ε with f r π arcin r inα+1 t dt. i If F E α C, then we have P α,φ F P α,φ E α 1ωh V φ E α F 3. ii If F E α C ε with <ε <1, then we have P α,φ F P α,φ E α ε 8[1 ε α+h +1 ε h ε]ω h V φ E α F. Here P α,φ Eup{ E div α x p ϕ dxdy : ϕ Cc 1Rh+1 ; R h+1, max ϕ 1} and V φ E E x p dxdy are called the weighted α-perimeter and the weighted volume of E, repectively. Finally ω h denote the Euclidean volume of the unit ball. When p intheorem1.1, we can obtain the quantitative ioperimetric inequalitie for the et E α in half-cylinder in Gruhin pace.

3 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 3 of 1 Preliminarie The Gruhin pace R h+1 {x, y :x R h, y R} i a Carnot-Carathéodory pace with a ytem of vector field X i x i, i 1,...,h and Y x α y, where α > i a given real number and x i the tandard Euclidean norm of x. The α-perimeter of a meaurable et E R h+1 in an open et A R h+1 i defined a { P α E; Aup E div α ϕ dxdy : ϕ Cc 1 A; R h+1, ϕ max ϕx, y } 1, x,y A where the α-divergence of the vector field ϕ : A R h+1 i given by div α ϕ X 1 ϕ X h ϕ h + Y ϕ h+1. If P α E; A<, by the Riez repreentation theorem there exit a poitive Radon meaure μ E on A and a μ E -meaurable function v E : A R h+1 uch that v E 1μ E -a.e. on A and the generalized Gau-Green formula div α ϕ dxdy ϕ, v E dμ E 3 E A hold for all ϕ C 1 c A; Rh+1. Here and hereafter,, denote the tandard Euclidean calar product. The meaure μ E i called α-perimeter meaure and the function v E i called meaure theoretic inner unit α-normal of E. NowweendowtheGruhinpaceR h+1 with denity e φ and define the weighted α- perimeter of a meaurable et E R h+1 in an open et A R h+1 a { P α,φ E; A up div α,φ ϕe φ dxdy : ϕ Cc 1 A; R h+1, E ϕ max ϕx, y } 1, 4 x,y A where div α,φ ϕ e φ div α e φ ϕ i called the weighted α-divergence of ϕ. By the definition of div α,φ ϕ,4 can alo be rewritten a { P α,φ E; A up div α e φ ϕ dxdy : ϕ Cc 1 A; R h+1, E ϕ max ϕx, y 1 }. 5 x,y A If P α,φ E; A<,thenby3wehave div α,φ ϕ dv φ ϕ, v E dμ E,φ, 6 E A where dv φ e φ dxdy i the weighted volume meaure and dμ E,φ e φ dμ E i called the weighted α-perimeter meaure. For any open et A R h+1,wehavep α,φ E; Aμ E,φ A. When A R h+1,letp α,φ EP α,φ E; R h+1.

4 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 4 of 1 Let be a hyperurface in the Gruhin pace R h+1 with denity e φ. can be locally given by the zero et of a function u C 1 uch that α u on, where α u X 1 u,...,x h u, Yu i called the α-gradient of u. ForaetE {x, y R h+1 : ux, y >}, the inner unit α-normal in equation 6igivenon E by the vector v E αux, y α ux, y. Then we define the weighted α-mean curvature of a H,φ 1 h div α,φ v E 1 h divα v E + v E, α φ. 7 Remark.1 Noticing that the α-mean curvature of i defined by H 1 h div α v E,then from 7wehave H,φ H 1 h v E, α φ. To prove Theorem 1.1, we need the following lemma. Lemma.1 Let the Gruhin pace R h+1 be endowed with denity e φ x p. For any ε <1,let C ε {x, y R h+1 : x <1,y > y ε } be half-cylinder, where y ε f 1 ε with f r π arcin r inα+1 t dt. There exit a continuou function u : C ε Rwithlevelet ε {x, y C ε : ux, y}, R, uch that i u C 1 C ε E α C 1 C ε \ E α and αux,y α ux,y i continuouly defined on C ε \{x }; ii >1 C ε E α and 1 C ε \ E α ; iii i a hyperurface of cla C with contant weighted α-mean curvature, that i, H,φ 1 1+ p h for >1 and H,φ 1+ p h for 1; iv for any point x, f x y with >1, we have 1 h H 1,φ x, f x y 5 y when ε 8 and 1 h H ε,φ x, f x y 1 ε α + y when <ε <1. 9 ε Proof The profile function of the et E α i the function f :[,1] R, π f r in α+1 t dt. 1 arcin r

5 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 5 of 1 It firt and econd derivative are f r rα+1, f r rα [αr α +1] r 1 r 3 We define the function g :[,1] R, grα +1f r rf r π α +1 in α+1 t dt + rα+. 1 arcin r 1 r It derivative i r α+1 g r 1 r 3 >. 13 Now we contruct a foliation of C ε.inc ε \ E α,theleave of the foliation are vertical tranlation of the top part of the boundary E α.inc ε E α,theleave are contructed a follow: the urface E α i dilated by a factor larger than 1 where dilation i defined by x, y λx, λ α+1 y λ >, and then it i tranlated downward in uch a way that the urface {y y ε f 1 ε} i alo the leaf at lat. We contruct a function u on the et C ε \ E α a ux, yf x y +1, x, y C ε \ E α. 14 Let {x, y C ε \ E α : ux, y }. Thenwehave 1and 1 E α.from14, we know u C 1 C ε \ E α and 1 C ε \ E α. In the following we will define the function u on the et D ε C ε E α. Setting r x and r ε 1 ε,weletf ε : D ε 1, R be a function F ε x, y, α+1 [f r f For any x, y D ε we have rε ] + y ε y. 15 lim F εx, y, f r y >, 1 + lim F ε x, y, y ε y <. On the other hand, uing 1and13wehave F ε α [g r g rε ] <. 16 So there exit a unique >1uchthatF ε x, y, for any x, y D ε.furthermorewe can define a function u : D ε R, ux, y determined by the equation F ε x, y,.obviouly we have u C 1 C ε E α andc ε E α >1,where {x, y C ε E α : ux, yideterminedbyf ε x, y, }. By 15, we find xi F ε x, y, α x i r f r, i 1,...,h; y F ε x, y, 1. 17

6 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 6 of 1 Uing 11, 16and17, we obtain xi ux, y x i r α α r [g r g r ε ], yux, y 1 α [g r g r ε ]. 18 Then we have X i u xi u x i r α α r [g r g r ε ], Yu r α x α y u α [g r g r ε ], and the quare length of the α-gradient of u on D ε i α u h X i u +Yu r α α r [g r g r ε ]. i1 Note that α u ifandonlyifx. So for any x, y D ε with x,wehave X i u α u x i, i 1,...,h; Yu α u r. 19 If x, y D ε tend to x, y E α with x andy >,then ux, y convergeto1. From 19, we have lim x,y x,y α ux, y α ux, y x 1,..., x h, 1 x αux, y α ux, y, where the right hand ide i computed by the definition 14ofu. The above equality how that αu α u i continuou on C ε \{x }. Inthecaeofe φ x p,wegetφ p ln x and α φ p x x 1,..., From 14, we know that the inner unit α-normal of with 1i v x 1,..., x h, 1 x. p x x h, for x. So the weighted α-mean curvature H,φ of with 1igivenby H,φ 1 h divα v Eα v Eα, α φ 1+ p h. From 19 we know that the inner unit α-normal of with >1i v x 1,..., x h, x So the weighted α-mean curvature H,φ of with >1igivenby H,φ 1 1+ p. h. Fixing a point x with x <1 ε and for y < f x y ε,wedefinethefunction h x yu x, f x y 1+ p 1, h H,φ

7 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 7 of 1 where 1 i uniquely determined by x, f x y. Then the function y h x y i increaing and h x 1. From 18and, we know h x y yu x, f x y 1 h x y α [g r ε h x y g r h x y ], for all y < f x y ε. By 13, g i trictly increaing. So h x yatifie h x y 1 h α x y[g r 1 ε h x g]. y On the other hand, for any >1wehave α [g rε ] rε g α α rε rε g r dr r α+1 dr 1 r 3 rε α r dr 1 r 3 [ 1 rε α rε ] 1 1 rε α rε r α ε 1 rε. When ε,wehaver ε 1.Soturninto α [g rε By 1, we get ] g 1. 1 h x y h x y 1, y. 3 Integrating 3withh x 1, we get h x y y. Thu we obtain 1 h H 1,φ x, f x y 1 h x y y 4+y.

8 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 8 of 1 Noticing we have y < f x π π f in α+1 t dt int dt 1, 1 h H 1,φ x, f x y 5 y. 4 When < ε <1,turninto α [g rε So by 1, we have h x y ] 1 εα g. 1+ε ε, 1 ε α y. 5 Integrating 5withh x 1, we have h x y 1+ ε 1 ε α y. Noticing y < f x y ε 1, o we have 1 h H 1,φ x, f x y 1 h x y ε 1 ε α y ε 1 ε α + y. 6 ε 3 Proof of Theorem 1.1 Let u : C ε R be the function given by Lemma.1 and let {x, y C ε : ux, y} be leave of the foliation, R. We define the vector field X : C ε \{x } R h+1 by X αu α u. Then X atifie the following propertie: i X 1; ii for x, y E α C ε,wehavexx, y v Eα x, y where v Eα x, y i the unit inner α-normal to E α ; iii for any point x, y with 1,wehave div α,φ Xx, y. 7

9 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 9 of 1 For any point x, y with >1,wehave div α,φ Xx, y 1 <. 8 Let F R h+1 be a et with finite weighted α-perimeter uch that V φ F V φ E α and F E α C ε. By Theorem.. in [7], without lo of generality we can aume that the boundary F of F i C. For δ >,leteα δ {x, y E α : x > δ}.by8and6, we have V φ E δ α \ F E δ α\f E δ α\f 1 x p dxdy div α,φ X x p dxdy { X, v F dμ F,φ F E δ α E δ α F X, v E δ α dμ E δ α,φ }. Letting δ + and uing the Cauchy-Schwarz inequality, we obtain V φ E α \ F 1 1 x p dxdy div α,φ X x p dxdy { X, v F dμ F,φ F E α { dμ Eα,φ F E α dμ F,φ X, v Eα dμ Eα,φ 1 { Pα,φ E α ; C ε \ F P α,φ F; E α }. 9 } } By a imilar computation, we alo have V φ F \ E α x p dxdy F\E α div α,φ X F\E α x p dxdy 1 { X, v F dμ F,φ + F\E α 1 { dμ F,φ F\E α E α F E α F dμ Eα,φ X, v Eα dμ Eα,φ } 1 { Pα,φ F; C ε \ E α P α,φ E α ; F }. 3 }

10 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 1 of 1 On the other hand, we have [ div α,φ X divα,φ X x p dxdy 1+ 1 V φ E α \ F From 9, 3and31, we obtain 1 { Pα,φ E α ; C ε \ F P α,φ F; E α } div α,φ X x p dxdy V φ E α \ F 1 div α,φ X V φ F \ E α 1 div α,φ X x p dxdy x p dxdy 1 { Pα,φ F; C ε \ E α P α,φ E α ; F } It i equivalent to ] x p dxdy 1 div α,φ X x p dxdy div α,φ X x p dxdy. P α,φ F P α,φ E α 1 div α,φ X x p dxdy. 3 For any x with x <1 ε, we define the vertical ection Eα x {y :x, y E α} and F x {y : x, y F}. By the Fubini theorem, we have 1 div α,φ X { x <1 ε} E x α\f x x p dxdy 1 div α,φ X x p dy dx. Letting mxl 1 Eα x \ Fx, where L 1 denote 1-dimenional Lebegue meaure, then we obtain 1 div α,φ X f x x p dxdy 1 div α,φ X { x <1 ε} f x mx { x <1 ε} mx 1 1 h x y where h x yux, f x y i the function introduced in. So from 3and33wehave P α,φ F P α,φ E α { x <1 ε} mx x p dy dx x p dy dx, x p dy dx. 34 h x y

11 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 11 of 1 When ε,by8 in Lemma.1 and the Hölder inequality, 34turn into P α,φ F P α,φ E α { x <1} mx 1 5 y dy x p dx 3 x mx 3p dx 15 { x <1} h 3 15ωh mx x p dx { x <1} 1ωh V φ E α F When < ε <1,by9 in Lemma.1, and the Hölder inequality, 34turninto mx ε P α,φ F P α,φ E α { x <1 ε} 1 ε α + ε ydy x p dx ε [1 ε α + x mx p dx ε] { x <1 ε} ε [1 ε α+h +1 ε h mx x p dx ε]ω h { x <1 ε} ε 8[1 ε α+h +1 ε h V φ E α F. 36 ε]ω h Acknowledgement Thi work i upported by the National Natural Science Foundation of China No ; No and the Natural Science Foundation of the Anhui Higher Education Intitution of China No. KJ17A34. The author would like to thank the referee for their valuable comment and uggetion which helped to improve the paper. Competing interet The author declare that they have no competing interet. Author contribution All author contributed equally and ignificantly in writing thi paper. All author read and approved the final manucript. Publiher Note Springer Nature remain neutral with regard to juridictional claim in publihed map and intitutional affiliation. Received: 18 February 17 Accepted: 6 June 17 Reference 1. Panu, P: Une inégalié iopérmétrique ur le groupe de Heienber. C. R. Math. Acad. Sci. 95, Panu, P: An ioperimetric inequality on the Heienberg group. Rend. Semin. Mat. Torino, Special Iue, Leonardi, GP, Manou, S: On the ioperimetric problem in the Heienberg group H n. Ann. Mat. Pura Appl. 1844, Danielli, D, Garofalo, N, Nhieu, D-M: A partial olution of the ioperimetric problem for the Heienberg group. Forum Math. 1, Ritoré, M, Roale, C: Area-tationary urface in the Heienberg group H 1.Adv.Math.19, Monti, R: Heienberg ioperimetric problem. The axial cae. Adv. Calc. Var. 11, Monti, R, Rickly, M: Convex ioperimetric et in the Heienberg group. Ann. Sc. Norm. Super. Pia, Cl. Sci. 85, Ritoré, M: A proof by calibration of an ioperimetric inequality in the Heienberg group H n. Calc. Var. Partial Differ. Equ. 441-, Garofalo, N, Nhieu, D-M: Ioperimetric and Sobolev inequalitie for Carnot-Carathéodory pace and the exitence of minimal urface. Commun. Pure Appl. Math. 49, Leonardi, GP, Rigot, S: Ioperimetric et on Carnot group. Hout. J. Math. 9, Monti, R, Morbidelli, D: Ioperimetric inequality in the Gruhin plane. J. Geom. Anal. 14,

12 Heand Zhao Journal of Inequalitie and Application 17 17:16 Page 1 of 1 1. Francechi, V, Monti, R: Ioperimetric problem in H-type group and Gruhin pace. Rev. Mat. Iberoam. 34, Morgan, F: Manifold with denity. Not. Am. Math. Soc. 58, Morgan, F: Manifold with denity and Perelman proof of the Poincaré conjecture. Am. Math. Mon. 116, Morgan, F, Pratelli, A: Exitence of ioperimetric region in R n with denity. Ann. Glob. Anal. Geom. 43, Roale, C, Cañete, A, Bayle, V, Morgan, F: On the ioperimetric problem in Euclidean pace with denity. Calc. Var. Partial Differ. Equ. 31, Betta, MF, Brock, F, Mercaldo, A, Poteraro, MR: Weighted ioperimetric inequalitie on R n and application to rearrangement. Math. Nachr. 814, Cianchi, A, Fuco, N, Maggi, F, Pratelli, A: On the ioperimetric deficit in Gau pace. Am. J. Math. 1331, Cabré, X, Ro-Oton, X, Serra, J: Euclidean ball olve ome ioperimetric problem with nonradial weight. C. R. Math. Acad. Sci. Pari 35, Cabré, X, Ro-Oton, X: Sobolev and ioperimetric inequalitie with monomial weight. J. Differ. Equ. 5511, Lu, G: Weighted Poincaré and Sobolev inequalitie for vector field atifying Hörmander condition and application. Rev. Mat. Iberoam. 83, Franchi, B, Lu, G, Wheeden, RL: Repreentation formula and weighted Poincaré inqualitie for Hörmander vector field. Ann. Int. Fourier Grenoble 45, Lu, G: Local and global interpolation inequalitie on the Folland-Stein Sobolev pace and polynomial on traighted group. Math. Re. Lett. 4, He, G, Zhao, P: The weighted ioperimetric-type and Sobolev-type inequalitie for hyperurface in Carnot group. Nonlinear Anal. 135, He, G, Zhao, P: The ioperimetric problem in the Gruhin pace R h+1 with denity x p.rend.semin.mat.univ.padova 17, to appear 6. Francechi, V, Leonardi, GP, Monti, R: Quantitative ioperimetric inequalitie in H n. Calc. Var. Partial Differ. Equ. 543, Franchi, B, Serapioni, R, Serra Caano, F: Meyer-Serrin type theorem and relaxation of variational integral depending on vector field. Hout. J. Math. 4,

TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL

GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National