关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

In the present chapter we consider the well-posedness of an abstract boundary-value problem for differential equations of elliptic type

$$- \upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}},\upsilon \left( T \right) = {{\upsilon }_{T}}$$

in an arbitrary Banach spaceEwith the positive operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on three points for the numerical solutions of this problem are presented. The well-posedness of these difference schemes in various Banach spaces are studied. The stability and coercive stability estimates in Holder norms for solutions of the high order of accuracy difference schemes of the mixed type boundary-value problems for elliptic equations are obtained.

Partial Differential Equations of Elliptic Type

This chapter is concerned with the introduction of the basic integral equations for two-dimensional elastic linear material problems. It starts by briefly reviewing the partial differential equations for linear elastic material and introducing the necessary notations involved in the formulation. These governing equations are also extended to deal with problems in which initial stress and strain type loads are applied. Such kind of loads are not only important to take into account temperature or other similar loads, but also to model nonlinear material behaviour when used in conjunction with a well established successive elastic solution technique.

Boundary Integral Equations

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Differential Forms With Applications To The Physical Sciences

This chapter focuses on the theory of integral equations. The chapter presents a systematic investigation of equations that are generally known as Fredholm integral equations of the second kind. For Fredholm integral equations, as for all linear equations, the theorem ϕ ( P ) = ϕ 0 ( P ) + ϕ *( P ) holds. Instead of this equation, a more general equation of the form, ϕ ( P ) = λ∂ D K ( P,P 1 ) ϕ(P) 1 dP 1 + f ( P ), the so-called equation with a parameter, is considered. Other restrictions, such as boundedness, continuity, and so on, are also highlighted in the chapter.

The theory of integral equations

Criterion for the Lp-dissipativity of second order differential operators with complex coefficients

We give complete algebraic characterizations of the L
 p
 -dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathop{\cal A}\nolimits}^{hk}(x)\partial_{k})$
 , where ${\mathop{\cal A}\nolimits}^{hk}(x)$
 are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L
 p
 -dissipative if and only if $$ \left({1\over 2}-{1\over p}\right)^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}}, $$
 ν being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the L
 p
 -dissipativity of the operator $\partial_{h} ({\mathop{\cal A}\nolimits}^{h}(x)\partial_{h})$
 , where ${\mathop{\cal A}\nolimits}^{h}(x)$
 are m× m matrices with complex L1loc entries, and we describe the maximum angle of L
 p
 -dissipativity for this operator. Keywords: L
 p
 -dissipativity, Algebraic conditions, Elasticity system Mathematics Subject Classification (2000): 47D03, 47D06, 47B44, 74B05

Criteria for the L p -dissipativity of systems of second order differential equations

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.

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

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.

Alberto Cialdea

Papers

Criterion for the Lp-dissipativity of second order differential operators with complex coefficients

Criteria for the L p -dissipativity of systems of second order differential equations

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

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

On an integral equation of the first kind arising in the theory of cosserat