Let B be the unit disc in R2,H be the completion of C0∞(B)under the norm||u||=(∫B|▽U|^2dx-∫Bu^2/(1-|x|^2)^2dx)^1/2,U∈C^∞0(B).By the method of blow-up analysis and an argument of rearrangement with respect to the...Let B be the unit disc in R2,H be the completion of C0∞(B)under the norm||u||=(∫B|▽U|^2dx-∫Bu^2/(1-|x|^2)^2dx)^1/2,U∈C^∞0(B).By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric dv=dx/(1-|x|^2)^2,we prove that,for any fixedα,0≤α<λp(B)-infu∈■,u≠0||U||^2■/||U||^2p,the supremum u∈■、||su||■≤1∫B^e^4π(1+α||su||^2p)U^2dx<+∞,■p>1.This is an analog of early results of Lu–Yang(Discrete Contin.Dyn.Syst.,2009)and Yang(Trans.Amer.Math.Soc.,2007),and extends those of Wang–Ye(Adv.Math.,2012)and Yang–Zhu(Ann.Global Anal.Geom.,2016).展开更多
Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser-Trudinger inequality on M. Furthermore, the constant we obtain is sharp.
In this paper,using the blow-up analysis,we prove a singular Hardy-Morser-Trudinger inequality,and find its extremal functions.Our results extend those of Wang and Ye(2012),Yang and Zhu(2016),Csato and Roy(2015)and Ya...In this paper,using the blow-up analysis,we prove a singular Hardy-Morser-Trudinger inequality,and find its extremal functions.Our results extend those of Wang and Ye(2012),Yang and Zhu(2016),Csato and Roy(2015)and Yang and Zhu(2017).展开更多
Let (M,g) be a compact Riemannian manifold without boundary, and (N,g) a compact Riemannian manifold with boundary. We will prove in this paper that the and can be attained. Our proof uses the blow-up analysis.
In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divi...In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.展开更多
We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplaci...We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].展开更多
We will show in this paper that if A is very close to 1, thenI（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dVcan be attained, where M is a compact-manifold ...We will show in this paper that if A is very close to 1, thenI（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dVcan be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.展开更多
In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trud...In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.展开更多
Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x...Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.展开更多
文摘Let B be the unit disc in R2,H be the completion of C0∞(B)under the norm||u||=(∫B|▽U|^2dx-∫Bu^2/(1-|x|^2)^2dx)^1/2,U∈C^∞0(B).By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric dv=dx/(1-|x|^2)^2,we prove that,for any fixedα,0≤α<λp(B)-infu∈■,u≠0||U||^2■/||U||^2p,the supremum u∈■、||su||■≤1∫B^e^4π(1+α||su||^2p)U^2dx<+∞,■p>1.This is an analog of early results of Lu–Yang(Discrete Contin.Dyn.Syst.,2009)and Yang(Trans.Amer.Math.Soc.,2007),and extends those of Wang–Ye(Adv.Math.,2012)and Yang–Zhu(Ann.Global Anal.Geom.,2016).
基金Supported by National Natural Science Foundation of China(Grant No.11201346)
文摘Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser-Trudinger inequality on M. Furthermore, the constant we obtain is sharp.
文摘In this paper,using the blow-up analysis,we prove a singular Hardy-Morser-Trudinger inequality,and find its extremal functions.Our results extend those of Wang and Ye(2012),Yang and Zhu(2016),Csato and Roy(2015)and Yang and Zhu(2017).
文摘Let (M,g) be a compact Riemannian manifold without boundary, and (N,g) a compact Riemannian manifold with boundary. We will prove in this paper that the and can be attained. Our proof uses the blow-up analysis.
基金supported by National Natural Science Foundation of China (Grant No. 11401575)
文摘In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.
文摘We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].
文摘We will show in this paper that if A is very close to 1, thenI（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dVcan be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.
基金Supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China。
文摘In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.
基金Supported by the National Natural Science Foundation of China(Grant No.11201346)。
文摘Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.
基金The Natural Science Foundation of NNSF funds of NNSF of China(11526177)the Natural Science Foundation of Jiangsu Province(BK20151160)the Nurturing Project of Xuzhou University of Technology(XKY2019103)
