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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Eigenfunction of the Laplace operator in +1-dimentional simplex

Vestnik MGSU 11/2014
  • Ovchintsev Mikhail Petrovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Sitnikova Elena Georgievna - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), .

Pages 68-73

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.

DOI: 10.22227/1997-0935.2014.11.68-73

References
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