Partial expansion of a Lipschitz domain and some applications
Partial expansion of a Lipschitz domain and some applications
摘要
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.
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.
关键词
Lipschitz
domain
regular
decomposition
mixed
boundary
condition
transversal
vector
field
extension
operator
Schwarz
preconditioner
bounded
cochain
projector
divergence
CURL
SchSberl
projector
Lipschitz ,domain, regular decomposition, mixed boundary condition, transversal vector field, extension operator, Schwarz preconditioner, bounded cochain projector, divergence, curl, SchSberl projector
参考文献31
-
1Amrouche C,Bernardi C,Dauge M,Girault V. Vector potentials in three-dimensional non-smooth domains[J].Mathematical Methods in the Applied Sciences,1998,(9):823-864.doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.
-
2Arnold D N,Falk R S,Winther R. Preconditioning in H(div) and applications[J].Mathematics of Computation,1997.957-984.
-
3Arnold D N,Falk R S,Winther R. Finite element exterior calculus:from Hodge theory to numerical stability[J].Bulletin of the American Mathematical Society(New Series),2010.281-353.
-
4Birman M,Solomyak M. Construction in a piecewise smooth domain of a function of the class H2 from the value of the conormal derivative[J].J Math Sov,1990.1128-1136.
-
5Bonnet-Ben Dhia A-S,Hazard C,Lohrengel S. A singular field method for the solution of Maxwell's equations in polyhedral domains[J].SIAM Journal of Applied Mathematics,1999.2028-2044.
-
6Bramble J H. A proof of the inf-sup condition for the Stokes equations on Lipschitz domains[J].Mathematical Methods in the Applied Sciences,2003.361-371.
-
7Brezzi F,Fortin M. Mixed and Hybrid Finite Element Methods[A].New York:springer-verlag,1991.
-
8Buffa A,Costabel M,Sheen D. On traces for H(curl,Ω) in Lipschitz domains[J].Journal of Mathematical Analysis and Applications,2002.845-867.
-
9Christiansen S H,Winther R. Smoothed projections in finite element exterior calculus[J].Mathematics of Computation,2008.813-829.
-
10Clément Ph. Approximation by finite element functions using local regularization[J].RAIRO Analyse Numérique,1975,(9e année):77-84.
-
1莫嘉琪,刘其林.具有非局部边界条件的奇摄动反应扩散问题[J].Journal of Mathematical Research and Exposition,1997,17(3):451-454. 被引量：3
-
2莫嘉琪.具有非局部边界条件的半线性椭圆型方程奇摄动问题[J].安徽师大学报,1996,19(2):101-106. 被引量：1
-
3L^p boundary value problems for Schr dinger equations in Lipschitz domain[J].Chinese Science Bulletin,1998,43(1):81-83.
-
4Zeng Jian LOU,Shou Zhi YANG,Dao Jin SONG.Decompositions of the Hardy space _z^1(Ω)[J].Acta Mathematica Sinica,English Series,2005,21(4):949-954.
-
5何萍.关于Lipschitz域上反射Brown运动的游程[J].数学年刊（A辑）,2006,27(2):259-268.
-
6何文平,周杰荣.带有非局部边界条件的反应——扩散方程解的爆破与整体存在[J].南通大学学报（自然科学版）,2007,6(1):4-7. 被引量：1
-
7谌跃中,王远弟.非局部边界条件的特征问题及发展问题[J].应用数学,2004,17(2):203-209. 被引量：2
-
8刘孝艳,冯象初,赵晨萍.Sobolev广义度量下的各向异性扩散模型[J].自动化学报,2015,41(2):320-329. 被引量：2
-
9何鹏,陶建华.基于Sobolev空间序列特征值问题的自然图像小尺度模式分析[J].自动化学报,2009,35(12):1568-1573.
-
10叶培新.QUANTUM COMPLEXITY OF SOBOLEV IMBEDDINGS[J].Acta Mathematica Scientia,2012,32(3):1102-1114.
