Diversity of Kirchhoff Transmission Conditions in Problems of Mathematical Physics

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Abstract
Several problems of mathematical physics for an angular junction of two thin rectangles, for example, beams, plates, or channels, are considered. Relying on an asymptotic analysis of all problems, we derive their one-dimensional models involving transmission conditions at the corner point where the segments, that is, the midlines of the rectangles with derived limiting ordinary differential equations of the models, meet each other. Among the obtained transmission conditions, we detect classical, weighted, and modified Kirchhoff ones, but the transmission conditions in the elasticity problem of deformation of an angular beam turns out to be nonstandard, namely, constrained due to the Dirichlet conditions for transverse displacements. The reasons for the formation of various structures of transmission conditions and their relationship with the energy functional of the original problem are explained. Miscellaneous examples are given.

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