Let ? be a bounded and connected open subset of R^N with a Lipschitzcontinuous boundary,the set ? being locally on the same side of ??.A vector version of a fundamental lemma of J.L.Lions,due to C.Amrouche,the first a...Let ? be a bounded and connected open subset of R^N with a Lipschitzcontinuous boundary,the set ? being locally on the same side of ??.A vector version of a fundamental lemma of J.L.Lions,due to C.Amrouche,the first author,L.Gratie and S.Kesavan,asserts that any vector field v =(vi) ∈(D′(?))~N,such that all the components 1/2(?_jv_i + ?_iv_j),1 ≤ i,j ≤ N,of its symmetrized gradient matrix field are in the space H^(-1)(?),is in effect in the space(L^2(?))~N.The objective of this paper is to show that this vector version of J.L.Lions lemma is equivalent to a certain number of other properties of interest by themselves.These include in particular a vector version of a well-known inequality due to J.Neˇcas,weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field,or a natural vector version of a fundamental surjectivity property of the divergence operator.展开更多
基金supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region,China(No.9041738-CityU 100612)
文摘Let ? be a bounded and connected open subset of R^N with a Lipschitzcontinuous boundary,the set ? being locally on the same side of ??.A vector version of a fundamental lemma of J.L.Lions,due to C.Amrouche,the first author,L.Gratie and S.Kesavan,asserts that any vector field v =(vi) ∈(D′(?))~N,such that all the components 1/2(?_jv_i + ?_iv_j),1 ≤ i,j ≤ N,of its symmetrized gradient matrix field are in the space H^(-1)(?),is in effect in the space(L^2(?))~N.The objective of this paper is to show that this vector version of J.L.Lions lemma is equivalent to a certain number of other properties of interest by themselves.These include in particular a vector version of a well-known inequality due to J.Neˇcas,weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field,or a natural vector version of a fundamental surjectivity property of the divergence operator.
