An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions
Elsayed M. E. Zayed · Quarterly of Applied Mathematics · 2003
This paper deals with the very interesting problem concerning the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R 3 {R^3} . The trace of the heat semigroup θ ( t ) = ∑ v = 1 ∞ exp ( − t μ v ) \theta \left ( t \right ) = \sum olimits _{v = 1}^\infty {\exp \left ( - t{\mu _v} \right )} , where { μ v } v = 1 ∞ \left \{ {{\mu _v}} \right \}_{v = 1}^\infty are the eigenvalues of the negative Laplacian − ∇ 2 = − ∑ β = 1 3 ( ∂ ∂ x β ) 2 - { abla ^2} = - {\sum olimits _{\beta = 1}^3 {\left ( {\frac {\partial }{{\partial {x^\beta }}}} \right )} ^2} in the