Lord Kelvin’s method of images in semigroup theory

Abstract

We show that Lord Kelvin's method of images is a way to prove generation theorems for semigroups of operators. To this end we exhibit three examples: a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem The general ideaLord Kelvin's method of images is an ingenious way of solving problems involving boundary conditions, see e.g. [4,7,11,14,18,19,24] and references given there. The idea of employing the method to prove generation theorems for semigroups of operators goes apparently back to W. Feller who constructed the semigroup of the minimal (and the reflected) Brownian motion on R + by noting that the space of odd (even) functions is left invariant by the semigroup of the unrestricted Brownian motion on R, see [12], pp. 341-343, comp. [3]. These semigroups are generated by the one-dimensional Laplacian with Dirichlet and Neumann boundary conditions at the origin, respectively. A question whether a similar construction can be carried out also in the case of more general boundary conditions, seems to have been opened for Communicated

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

“…While the nature of the first and the last conditions is transparent, the other two require a comment. As already mentioned, together they imply stability condition (9) (which is a common assumption in convergence theorems), and in fact, our theorems remain true if we simply assume (9). However, for the sake of applications, it is more convenient to assume the two conditions discussed above.…”

On moments-preserving cosine families and semigroups in C[0, 1]

“…Interestingly, the moments-preserving cosine family, as restricted to the space of even functions is the same as the cosine family generated by the Laplace operator with Neumann boundary conditions. Moreover, 'the odd part' of the cosine family is isometrically isomorphic to the cosine family with Robin boundary condition investigated previously in [8].…”

“…In particular, it was shown there that the requirement that the first two moments (i.e., both moments of order 0 and 1) vanish, leads to well-posed wave and diffusion equations in the space of H −1 (T )-distributions of zero average, where H −1 (T ) is the dual space of In this paper, we present an alternative approach to such problems. Namely, using Lord Kelvin's method of images, shown recently to be a useful tool for proving generation theorems in [7,8], we construct, in a quite explicit way, a cosine family in C[0, 1], generated by a restriction of the Laplace operator and preserving the first two moments. As it turns out, the domain of this cosine family's generator is the space of twice continuously differentiable functions f ∈ C[0, 1] satisfying the boundary conditions…”

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We use the newly developed Kelvin's method of images [7,8] to show existence of a unique cosine family generated by a restriction of the Laplace operator in C[0, 1], that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions.

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A general theorem on generation of moments-preserving cosine families by Laplace operators in C[0,1]

“…Paper [5] presents a different approach, applicable apparently in a broader context: it shows that the recently developed Lord Kelvin's method of images [3,4] provides natural tools for constructing moments-preserving cosine families. In particular, the main theorem of [5] states that there is a unique cosine family generated by a Laplace operator in C [0,1] that preserves the moments of order zero and 1 (about 0).…”