Arnold, Douglas N.; Scott, L. Ridgway; Vogelius, Michael Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon. (English) Zbl 0702.35208 Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 15, No. 2, 169-192 (1988). The paper deals with the boundary value problem div U\(=f\) in \(\Omega \subset R^ 2\), \(U=g\) on \(\partial \Omega\), if \(\int_{\Omega}f=\int_{\partial \Omega}g\cdot \nu\) what is the necessary condition for the existence of a solution U. If \(\partial \Omega\) is smooth, then a simple construction of U is possible. This construction fails in the case of non-smooth \(\partial \Omega\). The main result of the paper is the existence of the bounded linear map which is the inverse operator with respect to the above Dirichlet boundary value problem on a polygonal domain \(\Omega\). Reviewer: I.Bock Cited in 53 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35J99 Elliptic equations and elliptic systems Keywords:divergence operator; Dirichlet boundary value problem; polygonal domain × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] R.A. Adams , Sobolev Spaces , Academic Press , New York, NY , 1975 . MR 450957 | Zbl 0314.46030 · Zbl 0314.46030 [2] D.N. Arnold - F. Brezzi - J. Douglas , PEERS: A new mixed finite element for plane elasticity , Japan J. Appl. Math ., 1 ( 1984 ), pp. 347 - 367 . MR 840802 | Zbl 0633.73074 · Zbl 0633.73074 · doi:10.1007/BF03167064 [3] P. Grisvard , Elliptic Problems in Nonsmooth Domains , Pitman , Marshfield, MA , 1985 . MR 775683 | Zbl 0695.35060 · Zbl 0695.35060 [4] G.N. Jakovlev , Traces of functions in the space Wl p on piecewise smooth surfaces , Mat. Sb. (N.S.) , 74 ( 1967 ). Math. USSR-Sb. , 3 ( 1967 ), pp. 481 - 497 . MR 220053 | Zbl 0174.44004 · Zbl 0174.44004 · doi:10.1070/SM1967v003n04ABEH002758 [5] J.L. Lions - E. Magenes , Non-homogeneous Boundary Value Problems and Applications, I , Springer-Verlag , New York - Heidelberg - Berlin , 1972 . Zbl 0223.35039 · Zbl 0223.35039 [6] M. Morley , A family of mixed elements for linear elasticity , Univ. of Chicago , PhD Thesis, 1986 . · Zbl 0604.73095 [7] J Nečas , Les Méthodes Directes en Théorie des Équations Elliptiques , Masson et Cie ., Paris , 1967 . MR 227584 · Zbl 1225.35003 [8] L.R. Scott - M. Vogelius , Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials , Math. Modelling and Numer. Anal. , 19 ( 1985 ), pp. 111 - 143 . Numdam | MR 813691 | Zbl 0608.65013 · Zbl 0608.65013 [9] K.T. Smith , A Primer of Modern Analysis , Springer-Verlag , New York - Heidelberg - Berlin , 1983 . MR 710655 | Zbl 0517.26001 · Zbl 0517.26001 [10] E.M. Stein , Singular Integrals and Differentiability Properties of Functions , Princeton Univ. Press , 1970 . MR 290095 | Zbl 0207.13501 · Zbl 0207.13501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.