Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions

Author: E. M. E. Zayed
Journal: Quart. Appl. Math. 61 (2003), 233-249
MSC: Primary 35P20; Secondary 35J40, 35R30
DOI: https://doi.org/10.1090/qam/1976367
MathSciNet review: MR1976367
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Abstract: 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}$. The trace of the heat semigroup $ \theta \left( t \right) = \sum\nolimits_{v = 1}^\infty {\exp \left( - t{\mu _v} \right)}$, where $ \left\{ {{\mu _v}} \right\}_{v = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - {\nabla ^2} = - {\sum\nolimits_{\beta = 1}^3 {\left( {\frac{\partial }{{\partial {x^\beta }}}} \right)} ^2}$ in the $ \left( {x^1}, {x^2}, {x^3} \right)$-space, is studied for a general multiply-connected bounded domain $ \Omega $ in $ {R^3}$ surrounding by simply connected bounded domains $ {\Omega _j}$ with smooth bounding surfaces $ {S_j}\left( j = 1,...,n \right)$, where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components $ S_i^* \left( i = 1 + {k_{j - 1}},...,{k_j} \right)$ of the bounding surfaces $ {S_j}$ is considered, such that $ {S_j} = \cup _{i = 1 + {k_{j - 1}}}^{{k_j}} S_i^*$, where $ {k_0} = 0$. Some applications of $ \theta \left( t \right)$ for an ideal gas enclosed in the multiply-connected bounded container $ \Omega $ with Robin boundary conditions are given. We show that the asymptotic expansion of $ \theta \left( t \right)$ for short-time $ t$ plays an important role in investigating the influence of the finite container $ \Omega $ on the thermodynamic quantities of an ideal gas.

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DOI: https://doi.org/10.1090/qam/1976367
Article copyright: © Copyright 2003 American Mathematical Society

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