On the curl operator and some characterizations of matrix fields in Lipschitz domains

Abstract As well-known, De Rham's Theorem is a classical way to characterize vector fields as the gradient of the scalar fields, it is a tool of great importance in the theory of fluids mechanic. The first aim of this paper is to provide a useful rotational version of this theorem to establish several results on boundary value problems in the field of electromagnetism. Gurtin presented the completeness of Beltrami's representation in smooth domains for smooth symmetric matrix. The extension of Beltrami's completeness was obtained by Geymonat and Krasucki when the domain Ω is only Lipschitz and for symmetric matrix in L s 2 ( Ω ) . Our second aim is to present new extensions of Beltrami's completeness results on Lipschitz domains, firstly in the case where the data are in D s ( Ω ) and secondly when the data are in W s m , r ( Ω ) . The third objective is to extend Saint-Venant's Theorem to the case of distributions.

[1]  P. G. Ciarlet,et al.  On a lemma of Jacques-Louis Lions and its relation to other fundamental results , 2015 .

[2]  Dennis DeTurck,et al.  Vector Calculus and the Topology of Domains in 3-Space , 2002, Am. Math. Mon..

[3]  R. Fosdick,et al.  A Stokes theorem for second-order tensor fields and its implications in continuum mechanics , 2005 .

[4]  M. L. Albeggiani.  Osservazioni sulla nota precedente , 1890 .

[5]  Martin Costabel,et al.  On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains , 2008, 0808.2614.

[6]  George Herrmann,et al.  Linear Theory of Elasticity , 2000 .

[7]  Philippe G. Ciarlet,et al.  Characterization of the kernel of the operator CURL CURL , 2007 .

[8]  P. Ciarlet,et al.  ANOTHER APPROACH TO LINEARIZED ELASTICITY AND A NEW PROOF OF KORN'S INEQUALITY , 2005 .

[9]  Giovanni P. Galdi,et al.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems , 2011 .

[10]  George Biddell Airy,et al.  IV. On the strains in the Interior of beams , 1863, Philosophical Transactions of the Royal Society of London.

[11]  J. Ball,et al.  Partial regularity and smooth topology-preserving approximations of rough domains , 2013, 1312.5156.

[12]  L. Schwartz.  Théorie des distributions , 1966 .

[13]  On the characterizations of matrix fields as linearized strain tensor fields , 2006 .

[14]  Giuseppe Geymonat,et al.  Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains , 2008 .

[15]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[16]  M. E. Bogovskii.  Solution of the first boundary value problem for the equation of continuity of an incompressible medium , 1979 .

[17]  Philippe G. Ciarlet,et al.  WEAK VECTOR AND SCALAR POTENTIALS: APPLICATIONS TO POINCARÉ'S THEOREM AND KORN'S INEQUALITY IN SOBOLEV SPACES WITH NEGATIVE EXPONENTS , 2010 .

[18]  J. Moreau.  Duality characterization of strain tensor distributions in an arbitrary open set , 1979 .

[19]  W. Borchers,et al.  On the equations rot v=g and div u=f with zero boundary conditions , 1990 .

[20]  Sorin Mardare.  On Poincaré and de Rham‘s theorems , 2008 .

[21]  Elliptic boundary-value problems in nonsmooth domains , 1995 .

[22]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[23]  Marius Mitrea,et al.  Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds , 2004 .

[24]  J. Maxwell,et al.  I.—On Reciprocal Figures, Frames, and Diagrams of Forces , 1870, Transactions of the Royal Society of Edinburgh.

[25]  Beltrami's solutions of general equilibrium equations in continuum mechanics , 2006 .