We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was establi...We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was established recently by Chang and Wang.Our proof uses an alternative and elementary argument.展开更多
We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators.In particular,we show that there exists a compactly supported nonnegative Sobolev function u_(0)that attains the infimum(which wil...We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators.In particular,we show that there exists a compactly supported nonnegative Sobolev function u_(0)that attains the infimum(which will be a positive real number)of the set{{∫∫(u>0)×(u>0)|u(x)-u(y)|^(2)/|x-y|^(n+2σ)dxdy:u∈^(σ)(R^(n)),∫R^(n)u^(2)=1,|{u>0}|≤1}.Unlike the corresponding problem for the usual fractional Laplacian,where the domain of the integration is R^(n)×R^(n),symmetrization techniques may not apply here.Our approach is instead based on the direct method and new a priori diameter estimates.We also present several remaining open questions concerning the regularity and shape of the minimizers,and the form of the Euler-Lagrange equations.展开更多
基金supported in part by NSFC 11501034,NSFC 11571019 and the key project NSFC 11631002.
文摘We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was established recently by Chang and Wang.Our proof uses an alternative and elementary argument.
基金supported by Hong Kong RGC grants ECS 26300716 and GRF 16302519partially supported by NSFC 11922104 and 11631002.
文摘We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators.In particular,we show that there exists a compactly supported nonnegative Sobolev function u_(0)that attains the infimum(which will be a positive real number)of the set{{∫∫(u>0)×(u>0)|u(x)-u(y)|^(2)/|x-y|^(n+2σ)dxdy:u∈^(σ)(R^(n)),∫R^(n)u^(2)=1,|{u>0}|≤1}.Unlike the corresponding problem for the usual fractional Laplacian,where the domain of the integration is R^(n)×R^(n),symmetrization techniques may not apply here.Our approach is instead based on the direct method and new a priori diameter estimates.We also present several remaining open questions concerning the regularity and shape of the minimizers,and the form of the Euler-Lagrange equations.
