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.
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
In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a ...In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.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.
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, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudin...In this paper, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudinger functional. Since the prob-lems involve more complicated norm and multiple singular points, not only we can＇tuse the symmetrization to deal with a one-dimensional inequality, but also the pro-cesses of the blow-up analysis become more delicate. In particular, the new inequalityis more general than that of [1, 2].展开更多
Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(...Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(n))is defined by||u||F=(∫_(R_(n)(F^(n)(↓△u)+|u|^(n)dx)^(1/n)In this paper,the following sharp anisotropic Moser-Trudinger inequality involving L^(n)norm u∈W^(1,n)^(SUP)(R^(n),||u||F≤1∫_(R^(n))Ф(λ_(n)|u|n/n-1(1+a||u||^(n)_(n)1/n-1)dx<+∞in the entire space R^(n)for any 0<a<1 is estabished,whereФ(t)=e^(t)-∑^(n-2)_(j=0)tj/j!,λ_(n)=n^(n/n-1)k_(n)1/n-1 and kn is the volume of the unit Wulf ball in Rn.It is also shown that the above supremum is infinity for all α≥1.Moreover,we prove the supremum is attained,that is,there exists a maximizer for the above supremum whenα>O is sufficiently small.展开更多
We establish sufficient conditions under which the quasilinear equation -div（｜△↓u｜^n-2△↓u）＋V（x）｜u｜^n-2u=f（x,u）/｜x｜^β＋εh（x） in R^n,has at least two nontrivial weak solutions in W^1,n（R^n） when ...We establish sufficient conditions under which the quasilinear equation -div（｜△↓u｜^n-2△↓u）＋V（x）｜u｜^n-2u=f（x,u）/｜x｜^β＋εh（x） in R^n,has at least two nontrivial weak solutions in W^1,n（R^n） when ε 〉 0 is small enough, 0 〈β 〈 n, V is a continuous potential, f（x,u） behaves like exp{γ｜u｜^n/（n-1） } as ｜u｜→∞ for some γ 〉 0 and h 0 belongs to the dual space of W^1,n （Rn）.展开更多
Let(X,d,μ)be a metric space with a Borel-measureμ,supposeμsatisfies the Ahlfors-regular condition,i.e.birs≤μu(Br(x))≤b2rs,VBr(x)CX,r>0,where bi,b2 are two positive constants and s is the volume growth exponen...Let(X,d,μ)be a metric space with a Borel-measureμ,supposeμsatisfies the Ahlfors-regular condition,i.e.birs≤μu(Br(x))≤b2rs,VBr(x)CX,r>0,where bi,b2 are two positive constants and s is the volume growth exponent.In this paper,we mainly study two things,one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2.The other is to study the generalized Moser-Trudinger inequality with a singular Weight.展开更多
In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder con...In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>1.Our proof uses various PDE techniques but not the pluripotential theory.展开更多
Let Ω■R^(2) be a smooth bounded domain with 0∈■Ω.In this paper,we prove that for anyβ∈(0,1),the supremum u∈W^(1,2(Ω)),∫_(Ω)^(sup) udx=0,∫_(Ω)|▽u|^(2)dx≤1∫_(Ω)e^(2π(1-β)u^(2))/|x|^(2β)dx is finite a...Let Ω■R^(2) be a smooth bounded domain with 0∈■Ω.In this paper,we prove that for anyβ∈(0,1),the supremum u∈W^(1,2(Ω)),∫_(Ω)^(sup) udx=0,∫_(Ω)|▽u|^(2)dx≤1∫_(Ω)e^(2π(1-β)u^(2))/|x|^(2β)dx is finite and can be attained.This partially generalizes a well-known work of Chang and Yang(1988)who have obtained the inequality whenβ=0.展开更多
Extending the previous work [1], we establish well-posedness results for a more general class of semilinear wave equations with exponential growth. First, we investigate the well-posedness in the energy space. Then, w...Extending the previous work [1], we establish well-posedness results for a more general class of semilinear wave equations with exponential growth. First, we investigate the well-posedness in the energy space. Then, we prove the propagation of the regularity in the Sobolev spaces HS（IR^2） with s 〉 1. Finally, an ill-posedness result is obtained in HS（IR^2） for s 〈 1.展开更多
In this paper,we consider the convergence of the generalized Kähler-Ricci flow with semi-positive twisted formθon Kähler manifold M.We give detailed proofs of the uniform Sobolev inequality and some uniform...In this paper,we consider the convergence of the generalized Kähler-Ricci flow with semi-positive twisted formθon Kähler manifold M.We give detailed proofs of the uniform Sobolev inequality and some uniform estimates for the metric potential and the generalized Ricci potential along the flow.Then assuming that there exists a generalized Kähler-Einstein metric,if the twisting formθis strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field,we prove that the generalized Kähler-Ricci flow must converge in C^(∞)topology to a generalized Kähler-Einstein metric exponentially fast,where we get the exponential decay without using the Futaki invariant.展开更多
In this paper,we investigate a singular Moser-Trudinger inequality involving L^(n) norm in the entire Euclidean space.The blow-up procedures are used for the maximizing sequence.Then we obtain the existence of extrema...In this paper,we investigate a singular Moser-Trudinger inequality involving L^(n) norm in the entire Euclidean space.The blow-up procedures are used for the maximizing sequence.Then we obtain the existence of extremal functions for this singular geometric inequality in whole space.In general,W^(1,n)(R^(n))→L^(q)(R^(n))is a continuous embedding but not compact.But in our case we can prove that W^(1,n)(R^(n))→L^(n)(R^(n))is a compact embedding.Combining the compact embedding W^(1,n)(R^(n))→Lq(R^(n),|x|^(−s)dx)for all q≥n and 0<s<n in[18],we establish the theorems for any 0≤α<1.展开更多
This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimens...This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods(in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality.In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally,a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.展开更多
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below.Our proof does not rely on the uniformization theorem and the Onofri inequality,thu...In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below.Our proof does not rely on the uniformization theorem and the Onofri inequality,thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow.Such an analytic approach also sheds light on how to obtain the boundedness for E1 energy in the study of general Kähler manifolds.展开更多
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.
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].
基金supported by China Postdoctoral Science Foundation (Grant No. BX2021015)supported by National Key R&D Program of China (Grant No. SQ2020YFA0712800)National Natural Science Foundation of China (Grant No. 11822101)。
文摘In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.
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）.
文摘In this paper, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudinger functional. Since the prob-lems involve more complicated norm and multiple singular points, not only we can＇tuse the symmetrization to deal with a one-dimensional inequality, but also the pro-cesses of the blow-up analysis become more delicate. In particular, the new inequalityis more general than that of [1, 2].
基金Supported by Natural Science Foundation of China(Grant Nos.11526212,11721101,11971026)Natural Science Foundation of Anhui Province(Grant No.1608085QA12)+1 种基金Natural Science Foundation of Education Committee of Anhui Province(Grant Nos.KJ2016A506,KJ2017A454)Excellent Young Talents Foundation of Anhui Province(Grant No.GXYQ2020049)。
文摘Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(n))is defined by||u||F=(∫_(R_(n)(F^(n)(↓△u)+|u|^(n)dx)^(1/n)In this paper,the following sharp anisotropic Moser-Trudinger inequality involving L^(n)norm u∈W^(1,n)^(SUP)(R^(n),||u||F≤1∫_(R^(n))Ф(λ_(n)|u|n/n-1(1+a||u||^(n)_(n)1/n-1)dx<+∞in the entire space R^(n)for any 0<a<1 is estabished,whereФ(t)=e^(t)-∑^(n-2)_(j=0)tj/j!,λ_(n)=n^(n/n-1)k_(n)1/n-1 and kn is the volume of the unit Wulf ball in Rn.It is also shown that the above supremum is infinity for all α≥1.Moreover,we prove the supremum is attained,that is,there exists a maximizer for the above supremum whenα>O is sufficiently small.
文摘We establish sufficient conditions under which the quasilinear equation -div（｜△↓u｜^n-2△↓u）＋V（x）｜u｜^n-2u=f（x,u）/｜x｜^β＋εh（x） in R^n,has at least two nontrivial weak solutions in W^1,n（R^n） when ε 〉 0 is small enough, 0 〈β 〈 n, V is a continuous potential, f（x,u） behaves like exp{γ｜u｜^n/（n-1） } as ｜u｜→∞ for some γ 〉 0 and h 0 belongs to the dual space of W^1,n （Rn）.
文摘Let(X,d,μ)be a metric space with a Borel-measureμ,supposeμsatisfies the Ahlfors-regular condition,i.e.birs≤μu(Br(x))≤b2rs,VBr(x)CX,r>0,where bi,b2 are two positive constants and s is the volume growth exponent.In this paper,we mainly study two things,one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2.The other is to study the generalized Moser-Trudinger inequality with a singular Weight.
文摘In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>1.Our proof uses various PDE techniques but not the pluripotential theory.
基金supported by National Natural Science Foundation of China(Grant Nos.11721101 and 11401575)。
文摘Let Ω■R^(2) be a smooth bounded domain with 0∈■Ω.In this paper,we prove that for anyβ∈(0,1),the supremum u∈W^(1,2(Ω)),∫_(Ω)^(sup) udx=0,∫_(Ω)|▽u|^(2)dx≤1∫_(Ω)e^(2π(1-β)u^(2))/|x|^(2β)dx is finite and can be attained.This partially generalizes a well-known work of Chang and Yang(1988)who have obtained the inequality whenβ=0.
文摘Extending the previous work [1], we establish well-posedness results for a more general class of semilinear wave equations with exponential growth. First, we investigate the well-posedness in the energy space. Then, we prove the propagation of the regularity in the Sobolev spaces HS（IR^2） with s 〉 1. Finally, an ill-posedness result is obtained in HS（IR^2） for s 〈 1.
基金The work was supported in part by NSF in China,No.11131007,the Hundred Talents Program of CAS and Zhejiang Provincial Natural Science Foundation of China,No.LY12A01028.
文摘In this paper,we consider the convergence of the generalized Kähler-Ricci flow with semi-positive twisted formθon Kähler manifold M.We give detailed proofs of the uniform Sobolev inequality and some uniform estimates for the metric potential and the generalized Ricci potential along the flow.Then assuming that there exists a generalized Kähler-Einstein metric,if the twisting formθis strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field,we prove that the generalized Kähler-Ricci flow must converge in C^(∞)topology to a generalized Kähler-Einstein metric exponentially fast,where we get the exponential decay without using the Futaki invariant.
文摘In this paper,we investigate a singular Moser-Trudinger inequality involving L^(n) norm in the entire Euclidean space.The blow-up procedures are used for the maximizing sequence.Then we obtain the existence of extremal functions for this singular geometric inequality in whole space.In general,W^(1,n)(R^(n))→L^(q)(R^(n))is a continuous embedding but not compact.But in our case we can prove that W^(1,n)(R^(n))→L^(n)(R^(n))is a compact embedding.Combining the compact embedding W^(1,n)(R^(n))→Lq(R^(n),|x|^(−s)dx)for all q≥n and 0<s<n in[18],we establish the theorems for any 0≤α<1.
基金supported by the Projects STAB and Kibord of the French National Research Agency(ANR)the Project No NAP of the French National Research Agency(ANR)the ECOS Project(No.C11E07)
文摘This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods(in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality.In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally,a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.
基金We thank Yuxiang Li for pointing out the proof of Lemma 3.5 in an early version is incomplete.We also thank the referee for the careful reviewing and comments.
文摘In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below.Our proof does not rely on the uniformization theorem and the Onofri inequality,thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow.Such an analytic approach also sheds light on how to obtain the boundedness for E1 energy in the study of general Kähler manifolds.
