In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the c...In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s+t(Lp0(Ω))→Bq1s(Lp1(Ω)), t〉max{d(1/p0-1/p1), 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s+t(Lp0(Ω)) is a Besov space defined on the bounded Lipschitz domain Ω ? Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.展开更多
In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div(A∨u) - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cyli...In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div(A∨u) - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation Эtu-div(A∨u) + Vu = 0, where the singular potential V belonging to the Kato-Fefferman- Phong's class. Some applications are also discussed.展开更多
Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>(Ω)={f∈L<sup>1</sup>(Ω): there is an F∈ B<sub>1</sub><sup>0.1</sup>(R<s...Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>(Ω)={f∈L<sup>1</sup>(Ω): there is an F∈ B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>) such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>(Ω)={f∈B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>): f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.展开更多
We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We a...We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard wector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.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
We give a decomposition of the Hardy space Hz^1(Ω) into "div-curl" quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1(Ω) into Jacobians det Du, u ∈ W0^1,2 (Ω,R^2) for Ω in R^2....We give a decomposition of the Hardy space Hz^1(Ω) into "div-curl" quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1(Ω) into Jacobians det Du, u ∈ W0^1,2 (Ω,R^2) for Ω in R^2. This partially answers a well-known open problem.展开更多
基金supported by Natural Science Foundation of Inner Mongolia(Grant No.2011MS0103)supported by National Natural Science Foundation of China(Grant No.10671019)
文摘In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s+t(Lp0(Ω))→Bq1s(Lp1(Ω)), t〉max{d(1/p0-1/p1), 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s+t(Lp0(Ω)) is a Besov space defined on the bounded Lipschitz domain Ω ? Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.
基金supported in part by the NNSF of China (10471069, 10771110)by NSF of Ningbo City (2009A610084)
文摘In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div(A∨u) - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation Эtu-div(A∨u) + Vu = 0, where the singular potential V belonging to the Kato-Fefferman- Phong's class. Some applications are also discussed.
基金This research was partially supported by the SEDFthe NNSF of China.
文摘Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>(Ω)={f∈L<sup>1</sup>(Ω): there is an F∈ B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>) such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>(Ω)={f∈B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>): f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.
文摘We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard wector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.
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supported by Australian Government through the Australian Research Council,NNSF of China(Grant No.10371069)NSF of Guangdong Province(Grant No.032038)
文摘We give a decomposition of the Hardy space Hz^1(Ω) into "div-curl" quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1(Ω) into Jacobians det Du, u ∈ W0^1,2 (Ω,R^2) for Ω in R^2. This partially answers a well-known open problem.