Principles of Advanced Mathematical Physics pp 99-124 | Cite as

# Some Problems Connected with the Laplacian

## Abstract

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.

## Keywords

Vibration 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## Preview

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