Vita Leonessa

On Bernstein–Schnabl Operators on the Unit Interval

In this paper we study the Bernstein-Schnabl operators associated with a continuous selection of Borel measures on the unit interval. We investigate their approximation properties by presenting several estimates of the rate of convergence in terms of suitable moduli of smoothness. We also study some shape preserving properties as well as the preservation of the convexity. Moreover we show that their iterates converge to a Markov semigroup whose generator is a degenerate second order elliptic differential operator on the unit interval. Qualitative properties of this semigroup are also investigated together with its asymptotic behaviour.

On the Dirichlet and the Neumann problems for Laplace equation in multiply connected domains

In the classical indirect methods, the Dirichlet and the Neumann problems for Laplace equation are solved by means of a double and a simple layer potential, respectively. In this article we propose a method for obtaining the solutions of these problems in a multiply connected bounded domain of ℝ n (n ≥ 2) using different integral representations. Namely, we solve the Dirichlet problem by means of a simple layer potential and the Neumann problem through a double layer potential. An application in the theory of conjugate differential forms is also presented.

On the Dirichlet Problem for the Stokes System in Multiply Connected Domains

The Dirichlet problem for the Stokes system in a multiply connected domain of is considered in the present paper. We give the necessary and sufficient conditions for the representability of the solution by means of a simple layer hydrodynamic potential, instead of the classical double layer hydrodynamic potential.

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussed

Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains

The present paper is concerned with an indirect method to solve the Dirichlet and the traction problems for Lamé system in a multiply connected bounded domain of ℝn, n ≥ 2. It hinges on the theory of reducible operators and on the theory of differential forms. Differently from the more usual approach, the solutions are sought in the form of a simple layer potential for the Dirichlet problem and a double layer potential for the traction problem.2000 Mathematics Subject Classification. 74B05; 35C15; 31A10; 31B10; 35J57.

On a generalization of Kantorovich operators on simplices and hypercubes

Abstract In this paper we introduce and study two new sequences of positive linear operators acting on the space of all Lebesgue integrable functions defined, respectively, on the N-dimensional hypercube and on the N-dimensional simplex (N ≥ 1). These operators represent a natural generalization to the multidimensional setting of the ones introduced in [Altomare and Leonessa, Mediterr. J. Math. 3: 363–382, 2006] and, in a particular case, they turn into the multidimensional Kantorovich operators on these frameworks. We study the approximation properties of such operators with respect both to the sup-norm and to the Lp -norm and we give some estimates of their rate of convergence by means of certain moduli of smoothness.

The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains of () with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

Differential Operators and Approximation Processes Generated by Markov Operators

In this survey paper we report some recent results concerning some classes of differential operators as well as some sequences of positive approximation processes which can be constructed by means of a given Markov operator, the main aim being to investigate whether these differential operators are generators of positive semigroups and whether the semigroups can be approximated by iterates of the approximation processes themselves. Among other things, this theory discloses several interesting applications by highlighting, in particular, the relationship among positive semigroups, initial-boundary value problems, approximation theory, and Markov processes and by offering a unifying approach to the study of diverse differential problems.