Some Problems Connected with the Laplacian
The Laplacian is in many respects of a more classical nature than many of the differential operators to be discussed in Chapters 10 and 11. One of the basic problems is to find the eigenfunctions u(x) of the equation ∇ 2 u + λu = 0 in a region Ω of n-dimensional space with the boundary condition w(x) = 0 on the boundary ∂Ω. For n = 2 that is the classical problem of a vibrating membrane. More generally, for both n = 2 and n = 3 the eigenfunctions and the variational methods that determine them are useful in problems of vibration, heat flow, electromagnetic fields, and hydrodynamic stability. That is the main subject of this chapter.
KeywordsVibration eigenfunctions in a bounded domain variational methods the Dirichlet integral the potential due to a given charge distribution Poisson’s equation convolutions the direct product Schwartz’s nuclear theorem the Cauchy-Riemann equations harmonic functions
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