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

Abstract

Abstract. We use the newly developed Lord Kelvin's method of images (Bobrowski in J Evol Equ 10(3): 663-675, 2010; Semigroup Forum 81(3):435-445, 2010) 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. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.

Well-posedness and spectral properties of heat and wave equations with non-local conditions

“…One may extend A V {0} 2 ,K to an operator defined on the whole L 2 (0, 1) by dropping the conditions µ 0 (u) = µ 1 (u) = 0 and keeping the above boundary conditions. This new operator is perhaps more natural and has been extensively studied in [4].…”

Some Remarks on the Krein-von Neumann Extension of Different Laplacians

“…cf. also [BM13]. As in [CM09, § 5], a direct application of the Beurling-Deny conditions then shows that the associated semigroup is neither positive nor L ∞contractive, since neither of these properties is enjoyed by the semigroup…”

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

“…First, following [5] we will relate preservation of moments with boundary conditions to show that a generator A ∈ L s of a semigroup that preserves two moments of order i, j ≥ 1 could not be densely defined (thus establishing (b)). Then, in the next section, we will construct the moments preserving cosine family of point (a).…”

“…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).…”

“…In this context, a natural question arises of whether and to what extent the main result of [5], may be generalized to the case of two arbitrary moments of order, say i and j , where i < j are two non-negative integers. Our main theorem (Theorem 2.1) provides the following answer to this question: a necessary and sufficient condition for existence of a cosine family generated by a Laplace operator in C[0, 1] that preserves the moments of order i and j (about 0) is that i = 0.…”