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: On the solvability conditions of the first boundary value problem for a system of elliptic equations that strongly degenerate at a point | Rutkauskas Boundary Value Problems 2011 2011 16 http content 2011 1 16 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access On the solvability conditions of the first boundary value problem for a system of elliptic equations that strongly degenerate at a point Stasys Rutkauskas Correspondence stasys. rutkauskas@ Institute of Mathematics and Informatics of Vilnius University Akademijos str. 4 LT-08863 Vilnius Lithuania Springer Abstract A system of elliptic equations which are irregularly degenerate at an inner point is considered in this article. The equations are weakly coupled by a matrix that has multiple zero eigenvalue and corresponding to it adjoint vectors. Two statements of a well-posed Dirichlet type problem in the class of smooth functions are given and sufficient conditions on the existence and uniqueness of the solutions are obtained. Keywords systems of elliptic equations degenerate elliptic equations boundary value problems Dirichlet type problem 1 Introduction and statement of the problems The first results in the area of boundary value problems for an elliptic equation with degeneracy at an inner point of the considered domain are obtained in 1 . In that study the Dirichlet problem for a weakly regularly degenerating elliptic equation with the main part of Laplace s operator is studied. These results are developed in 2 where the degenerate elliptic operator is generalized and over and above the second boundary value problem is investigated. In 3 the existence of a weak solution to the Dirichlet problem for an elliptic equation degenerating at isolated points in the class of Holder functions is proved. In the case of the strong irregular degeneracy can new effects emerge which influence the well-posedness of the boundary value problems. For instance in 4 it is shown that in a well-posed Dirichlet type problem the asymptotic of the solution near the degeneracy point is supposed to be known. Many
