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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: CLASSES OF ELLIPTIC MATRICES | CLASSES OF ELLIPTIC MATRICES ANTONIO TARSIA Received 12 December 2005 Revised 20 February 2006 Accepted 21 February 2006 The equivalence between some conditions concerning elliptic matrices is shown namely the Cordes condition a generalized form of Campanato s condition and a generalized form of a condition of Buica. Copyright 2006 Antonio Tarsia. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let o be an open bounded set in R n 2 with a sufficiently regular boundary and let A x aij x i j 1 . n be a real matrix with coefficients aij e L O . We consider the following problem u e H2 2 n H01 2 O _ 1.1 y aij x Diju x f x a.e. x e o. t j 1 If f e L2 O it is known see the counterexamples in 6 that problem 1.1 is not well posed with the only hypothesis of uniform ellipticity on the matrix A x there exists a positive constant V such that n X aij x piPj Vllnll a.e. in o Wp n1 . P e R . 1.2 t j 1 It is therefore essential in order to be able to solve Problem 1.1 to assume some hypotheses on A x stronger than 1.2 . In this paper we consider some of these ones and compare them. More precisely we will consider the following conditions and show that they are equivalent. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID74171 Pages 1-8 DOI 10.1155 JIA 2006 74171 2 Classes of elliptic matrices Condition 1.1 the Cordes condition see 5 8 . IIA .x II R 2 0 a.e. in Q and there exists G 0 1 such that w j 1 a i x 2 Ki 1 a1j x n - 1 a.e. in Q. 1.3 Condition 1.2 Condition Axp . There exist four real constants Ơ Y 8 p with Ơ 0 Y 0 8 0 Y 8 1 p 1 and a function a x G ư Q with a x Ơ a.e. in Q such that nn Y kii - a x aij x kij i 1 i j 1 p YllkIlp2 8 n p 1 ill i 1 1.4 for all k kij i j 1 . n G Rn2 a.e. in Q. When p 1 the above condition will be simply denoted by Condition Ax
