We consider large-time behaviors of weak solutions to the evolutionary p-Laplacian with logarithmic source of time-dependent coefficient.We find that the weak solutions may neither decay nor blow up,provided that the ...We consider large-time behaviors of weak solutions to the evolutionary p-Laplacian with logarithmic source of time-dependent coefficient.We find that the weak solutions may neither decay nor blow up,provided that the initial data u(·,t_(0))is on the Nehari manifold N:={v∈W_(0)^(1,p)(Ω):I(v,to)=0,||▽v||P^(P)≠0}.This is quite different from the known results that the weak solutions may blow up as,u(·,to)∈N^(+):={v∈W_(0)^(1,p)(Ω):I(v,t_(0))<0}and weak solutions may decay as u(·,t_(0))∈N^(+):={v∈W_(0)^(1,p)(Ω):I(v,t_(0))>0}.展开更多
Links and knots are exotic topological structures that have garnered significant interest across multiple branches of natural sciences.Coherent links and knots,such as those constructed by phase or polarization singul...Links and knots are exotic topological structures that have garnered significant interest across multiple branches of natural sciences.Coherent links and knots,such as those constructed by phase or polarization singularities of coherent light,have been observed in various three-dimensional optical settings.However,incoherent links and knots—knotted or connected lines of coherence singularities—arise from a fundamentally different concept.They are"hidden"in the statistic properties of a randomly fluctuating field,making their presence often elusive or undetectable.Here,we theoretically construct and experimentally demonstrate such topological entities of incoherent light.By leveraging a state-of-the-art incoherent modal-decomposition scheme,we unveil incoherent topological structures from fluctuating light speckles,including Hopf links and Trefoil knots of coherence singularities that are robust against coherence and intensity fluctuations.Our work is applicable to diverse wave systems where incoherence or practical coherence is prevalent,and may pave the way for design and implementation of statistically-shaped topological structures for various applications such as high-dimensional optical information encoding and optical communications.展开更多
The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular li...The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H^2.展开更多
文摘We consider large-time behaviors of weak solutions to the evolutionary p-Laplacian with logarithmic source of time-dependent coefficient.We find that the weak solutions may neither decay nor blow up,provided that the initial data u(·,t_(0))is on the Nehari manifold N:={v∈W_(0)^(1,p)(Ω):I(v,to)=0,||▽v||P^(P)≠0}.This is quite different from the known results that the weak solutions may blow up as,u(·,to)∈N^(+):={v∈W_(0)^(1,p)(Ω):I(v,t_(0))<0}and weak solutions may decay as u(·,t_(0))∈N^(+):={v∈W_(0)^(1,p)(Ω):I(v,t_(0))>0}.
基金supported by the National Key Research and Development Program of China(No.2022YFA1404800)National Natural Science Foundation of China(No.12174280,No.12204340,No.12192254,No.92250304,No.12434012,No.W2441005)Priority Academic Program Development of Jiangsu Higher Education Institutions,and Postgraduate Research&Practice Innovation Programof Jiangsu Province(KYCX24_3287).
文摘Links and knots are exotic topological structures that have garnered significant interest across multiple branches of natural sciences.Coherent links and knots,such as those constructed by phase or polarization singularities of coherent light,have been observed in various three-dimensional optical settings.However,incoherent links and knots—knotted or connected lines of coherence singularities—arise from a fundamentally different concept.They are"hidden"in the statistic properties of a randomly fluctuating field,making their presence often elusive or undetectable.Here,we theoretically construct and experimentally demonstrate such topological entities of incoherent light.By leveraging a state-of-the-art incoherent modal-decomposition scheme,we unveil incoherent topological structures from fluctuating light speckles,including Hopf links and Trefoil knots of coherence singularities that are robust against coherence and intensity fluctuations.Our work is applicable to diverse wave systems where incoherence or practical coherence is prevalent,and may pave the way for design and implementation of statistically-shaped topological structures for various applications such as high-dimensional optical information encoding and optical communications.
文摘The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H^2.
