摘要:

AbstractGiven a probability space (X, μ) and a bounded domain Ω in ℝdequipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions onXand Ω) that for every vector‐valued function u ∈ Lp(X, μ; ℝd) there is a unique “polar factorization”u= ∇Ψs, where Ψ is a convex function defined on Ω andsis a measure‐preserving mapping from (X, μ) into (Ω, |·|), provided thatuis nondegenerate, in the sense that μ(u−1(E)) = 0 for each Lebesgue negligible subsetEof ℝd.Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real‐valued functions are unified.The Monge‐Ampère equation is involved in the polar factorization and the proof relies on the study of an ap

ISSN:0010-3640    

DOI:10.1002/cpa.3160440402

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

摘要:

AbstractIn this paper, we present a new numerical method for the solution of linear two‐point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system usingO(N·p2) operations, whereNis the number of nodes on the interval andpis the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end‐point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element me

ISSN:0010-3640    

DOI:10.1002/cpa.3160440403

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

摘要:

AbstractEnergy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects.

ISSN:0010-3640    

DOI:10.1002/cpa.3160440404

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0010-3640    

DOI:10.1002/cpa.3160440405

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY

ISSN:0010-3640    

DOI:10.1002/cpa.3160440401

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1991

数据来源: WILEY