In this article, the unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations are discussed. It is shown that strong solutions of the 2-component Degasperis-Procesi e...In this article, the unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations are discussed. It is shown that strong solutions of the 2-component Degasperis-Procesi equations, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if they also decay exponentially at a later time.展开更多
In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are ...In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.展开更多
Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero...Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.展开更多
Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato-Fefferman-Phong's class in Lipschitz domains. An elementary proof of the doubling property for u^2 over balls is pres...Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato-Fefferman-Phong's class in Lipschitz domains. An elementary proof of the doubling property for u^2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the Bp weight properties for the solution u near the boundary.展开更多
The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic fun...The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.The nu-merical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.展开更多
Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies ...Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies a three cylinder inequality near Γ x (—T, T). As aconsequence of the previous result we prove that if u(x,t) = O (|x|~k ) for every t ∈ (-T,T) andevery k ∈ N, then u is identically equal to zero.展开更多
In this paper, properties of the spherical functions and Hardy-Sobolev inequalities of generalized Baouendi-Grushin vector fields are established, and then some unique continuation results for generalized Baouendi Gru...In this paper, properties of the spherical functions and Hardy-Sobolev inequalities of generalized Baouendi-Grushin vector fields are established, and then some unique continuation results for generalized Baouendi Grushin operators with singular weights are given.展开更多
In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities wi...In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.展开更多
Chinese civilization is the only ancient civilization thriving to this day seems to be a common view in the historical academia and Chinese society as well. Culture is a way that human beings actively adapt to the sur...Chinese civilization is the only ancient civilization thriving to this day seems to be a common view in the historical academia and Chinese society as well. Culture is a way that human beings actively adapt to the surroundings and keep consistence with environmental change. All cultures created by human beings are changing, so is the Chinese culture. The physical and cultural evolution presented by the archaeological discoveries and related historical researches show that Chinese culture is multi-sourced and multicultural. This is the secret for the Chinese culture to retain its vitality. The reasons that Chinese civilization is the only civilization thriving to this day among all of ancient civilizations in the world include the ancestral worship, the identity to common ancestors, traditions of writing pedigrees and history books, the longhistory Chinese character system and the creative thinking and cultural methods of interpreting new thoughts by using traditional Chinese thinking and cultural structures. Especially in modern times, impacted by the nationalism trend, the national state narration built for meeting the requirement of the national state"historical memory"left us the strong impression of Chinese civilization being the only continued civilization and fostered the affective identification with the common history.展开更多
The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper.First,a reciprocity relation which involves two processes at different instants is established to form the basis of a u...The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper.First,a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem.The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion.The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients.Then the continuous dependence theorem is discussed upon two external data systems.Finally,the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem(mixed problem) is obtained.These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.展开更多
基金supported by NNSFC(11001219,10925104)the Scientific Research Program Funded by Shaanxi Provincial Education Department(2010JK860)
文摘In this article, the unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations are discussed. It is shown that strong solutions of the 2-component Degasperis-Procesi equations, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if they also decay exponentially at a later time.
基金supported by the National Natural Science Foundation of China(10871157)Research Fund for the Doctoral Program of Higher Education of China(200806990032)Keji Chuangxin Jijin of Northwestern Polytechnical University(2007KJ01012)
文摘In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.
文摘Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.
基金supported by National Nature of Science Foundation of China(No.10471069)by Natural Science Foundation of Zhejiang province of China(No.102066)by NSF of Ningbo city(No.2006A610090)
文摘Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato-Fefferman-Phong's class in Lipschitz domains. An elementary proof of the doubling property for u^2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the Bp weight properties for the solution u near the boundary.
基金This work was supported by the National Natural Science Foundation of China(No.11971121).
文摘The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.The nu-merical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.
基金This work is partially supported by MURST,Grant No.MM01111258
文摘Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies a three cylinder inequality near Γ x (—T, T). As aconsequence of the previous result we prove that if u(x,t) = O (|x|~k ) for every t ∈ (-T,T) andevery k ∈ N, then u is identically equal to zero.
基金Supported by National Natural Science Foundation of China (Grant No. 10871157), Research Fund for the Doctoral Program o[ Higher Education of China (Grant No. 200806990032)
文摘In this paper, properties of the spherical functions and Hardy-Sobolev inequalities of generalized Baouendi-Grushin vector fields are established, and then some unique continuation results for generalized Baouendi Grushin operators with singular weights are given.
基金supported by National Natural Science Foundation of China(Grant No.11401310)supported by National Natural Science Foundation of China(Grant No.11531005).
文摘In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.
文摘Chinese civilization is the only ancient civilization thriving to this day seems to be a common view in the historical academia and Chinese society as well. Culture is a way that human beings actively adapt to the surroundings and keep consistence with environmental change. All cultures created by human beings are changing, so is the Chinese culture. The physical and cultural evolution presented by the archaeological discoveries and related historical researches show that Chinese culture is multi-sourced and multicultural. This is the secret for the Chinese culture to retain its vitality. The reasons that Chinese civilization is the only civilization thriving to this day among all of ancient civilizations in the world include the ancestral worship, the identity to common ancestors, traditions of writing pedigrees and history books, the longhistory Chinese character system and the creative thinking and cultural methods of interpreting new thoughts by using traditional Chinese thinking and cultural structures. Especially in modern times, impacted by the nationalism trend, the national state narration built for meeting the requirement of the national state"historical memory"left us the strong impression of Chinese civilization being the only continued civilization and fostered the affective identification with the common history.
基金Project supported by the National Natural Science Fundation of China(Nos.11572358 and 11272223)the Training Program for Leading Talent in University Innovative Research Team in Hebei Province(No.LJRC006)
文摘The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper.First,a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem.The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion.The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients.Then the continuous dependence theorem is discussed upon two external data systems.Finally,the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem(mixed problem) is obtained.These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.