Собственная функция оператора Лапласа в +1-мерном симплексе

Abstract

In order to find eigenfunction of the Laplace operator in regular n+1-dimensional simplex the barycentric coordinates are used. For obtaining this result we need some formulas of the analytical geometry. A similar result was obtained in the earlier papers of the author in a tetrahedron from R 3 and in gipertetrahedron from R 4. Let П be unlimited cylinder in the space R n, its cross-section with hyperplane has a special form. Let L be a second order linear differential operator in divergence form, which is uniformly elliptic and η is its ellipticity constant. Let u be a solution of the mixed boundary value problem in Π with homogeneous Dirichlet and Neumann data on the boundary of the cylinder. In some cases the eigenfunction of the Laplace operator allows us to continue this solution from the cylinder Π to the whole space R n with the same ellipticity constant. The obtained result allows us to get a number of various theorems on the solution growth for mixed boundary value problem for linear differential uniformly elliptical equation of the second order, given in unlimited cylinder with special cross-section. In addition we consider n-1-dimensional hill tetrahedron and the eigenfunction for an elliptic operator with constant coefficients in it.