Some historical remarks on the positivity of boundary integral operators

Martin Costabel

Preprint IRMAR No 06-20 Université de Rennes 1 (2006) 27 p.

Ch. 1 of Boundary Element Analysis - Mathematical Aspects and Applications (M. Schanz, O. Steinbach, Eds.)
Lecture Notes in Applied and Computational Mechanics. Vol 29, 1-27. Springer, Berlin 2007.

In the analysis of boundary integral equations, energy methods play an important role. More precisely, these are variational methods based on positive definite quadratic forms that have an interpretation as energy functionals. They allow to obtain not only existence and uniqueness results for the integral equations, but also stability and convergence estimates for numerical approximations by the boundary element method. They work in very similar general ways for applications in many different models of mathematical physics, from classical potential theory and elasticity theory -- isotropic and anisotropic -- to fluid dynamics, electrodynamics, transient heat conduction and even models described by boundary value problems for higher order partial differential equations, like plate problems. Finally, they require much less regularity of the boundary of the domain than the classical theory of Fredholm integral equations of the second kind or methods based on local Fourier analysis. They also have a long history, going back to the early 19th century, with long periods where they were virtually ignored, until reappearing relatively recently and sometimes surprisingly.

Whereas the importance of these energy methods for first kind boundary integral equations has been appreciated for more than 30 years, their application to second kind integral equations has only rather recently led to important progress, and there remain many paths to explore. In the paper, we present some aspects of the curious history of the energy methods for boundary integral equations of the first and second kind. We discuss in particular Gauss' missing theorem from 1839 and famous but forgotten Poincaré inequalities from 1895. The latter lead to some new results about the convergence of Neumann's series in the energy norm.

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