In this paper,we consider the equivalent conditions with L^(p)-version(1<p<∞)of the J.L.Lions lemma.As applications,we first derive the existence of a weak solution to the Maxwell-Stokes type problem and then w...In this paper,we consider the equivalent conditions with L^(p)-version(1<p<∞)of the J.L.Lions lemma.As applications,we first derive the existence of a weak solution to the Maxwell-Stokes type problem and then we consider the Korn inequality.Furthermore,we consider the relation to other fundamental results.展开更多
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
The author first reviews the classical Korn inequality and its proof.Following recent works of S.Kesavan,P.Ciarlet,Jr.,and the author,it is shown how the Korn inequality can be recovered by an entirely different proof...The author first reviews the classical Korn inequality and its proof.Following recent works of S.Kesavan,P.Ciarlet,Jr.,and the author,it is shown how the Korn inequality can be recovered by an entirely different proof.This new proof hinges on appropriate weak versions of the classical Poincar'e and Saint-Venant lemma.In fine,both proofs essentially depend on a crucial lemma of J.L.Lions,recalled at the beginning of this paper.展开更多
文摘In this paper,we consider the equivalent conditions with L^(p)-version(1<p<∞)of the J.L.Lions lemma.As applications,we first derive the existence of a weak solution to the Maxwell-Stokes type problem and then we consider the Korn inequality.Furthermore,we consider the relation to other fundamental results.
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.
文摘The author first reviews the classical Korn inequality and its proof.Following recent works of S.Kesavan,P.Ciarlet,Jr.,and the author,it is shown how the Korn inequality can be recovered by an entirely different proof.This new proof hinges on appropriate weak versions of the classical Poincar'e and Saint-Venant lemma.In fine,both proofs essentially depend on a crucial lemma of J.L.Lions,recalled at the beginning of this paper.
