Multiplicative perturbations of the Laplacian and related approximation problems

Abstract

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on R N , N ≥ 1. It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C 0 -semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on R N . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.

Multivariate approximation by modified Gauss-Weierstrass operators

“…Proof We only need to verify the assumptions of Theorem 3.5 in [3] (see also Theorem 4.3 in [9]) and Lemma 2.1. Indeed, as n → ∞…”

“…The coefficients a i j , b i and c are arbitrary real continuous functions on R d , thus including the results of [6,8,9].…”

“…These operators have been used in [9] to approximate and investigate Feller semigroups generated by bounded multiplicative perturbations of the Laplacian.…”

Francesco Altomare - the remarkable mathematician and human being

“…Moreover Francesco, together with some collaborators, studied semigroups of positive operators also in connection with elliptic boundary value problems and Markov processes, especially in the setting of weighted continuous function spaces. We refer, e.g., to [25] and [26] and the references therein, for more details in this respect. He thinks that the devil lies in the details, so he approaches every Mathematical problem with rigor and care.…”

Parabolic variational inequalities with generalized reflecting directions

“…While our generalization is focused on transforming the reflection direction at the frontier of the domain into a new one, which is no longer normal at the frontier, different research directions aim at perturbing the Laplacian operator (see Eidus [8], Barbu, Favini [9] or Altamore, Milella, Musceo [10]). Multiplicative perturbations of the Laplacian play an important role in the theory of wave propagation in nonhomogeneous media whose density is related to the perturbation coefficient.…”