Alberto Cialdea

ON AN INTEGRAL EQUATION OF THE FIRST KIND ARISING IN THE THEORY OF COSSERAT

In this paper we study an integral equation of the first kind concerning an indirect boundary integral method for the Dirichlet problem in the theory of Cosserat continuum. Our method hinges on the theory of reducible operators and on the theory of differential forms.

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.

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.

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.

The -dissipativity of first order partial differential operators

We find necessary and sufficient conditions for the -dissipativity of the Dirichlet problem for systems of partial differential operators of the first order with complex locally integrable coefficients. As a by-product we obtain sufficient conditions for a certain class of systems of the second order.

L^p-dissipativity for Systems of Partial Differential Operators

This chapter is devoted to systems of partial differential operators. After some auxiliary results in Section 4.1, we give an algebraic necessary condition for the \(L^p\)-dissipativity of a general system in the two-dimensional case (Section 4.2). Several results are stated in terms of eigenvalues of the coefficient matrix of the system.

The Angle of L^p-dissipativity

The analyticity of a contractive semigroup \(\left\{T(t)\right\}\) is closely connected with the possibility of extending \(\left\{T(t)\right\}\) to a contractive semigroup \(\left\{T(z)\right\}(z\;\in\;\mathbb{C})\) in an angle, called angle of dissipativity.

Weighted Positivity and Other Related Results

Most of the results in the present chapter concern the \(L^2\)-weighted positivity of different operators. In the case of functions taking scalar values, by this positivity we mean the inequality