Search results for "Cauchy's integral formula"

Regularity of solutions of cauchy problems with smooth cauchy data

1988

The Cauchy problem for linear growth functionals

2003

In this paper we are interested in the Cauchy problem $$ \left\{ \begin{gathered} \frac{{\partial u}}{{\partial t}} = div a (x, Du) in Q = (0,\infty ) x {\mathbb{R}^{{N }}} \hfill \\ u (0,x) = {u_{0}}(x) in x \in {\mathbb{R}^{N}}, \hfill \\ \end{gathered} \right. $$ (1.1) where \( {u_{0}} \in L_{{loc}}^{1}({\mathbb{R}^{N}}) \) and \( a(x,\xi ) = {\nabla _{\xi }}f(x,\xi ),f:{\mathbb{R}^{N}}x {\mathbb{R}^{N}} \to \mathbb{R} \)being a function with linear growth as ‖ξ‖ satisfying some additional assumptions we shall precise below. An example of function f(x, ξ) covered by our results is the nonparametric area integrand \( f(x,\xi ) = \sqrt {{1 + {{\left\| \xi \right\|}^{2}}}} \); in this case …

Integral holomorphic functions

2004

We define the class of integral holomorphic functions over Banach spaces; these are functions admitting an integral representation akin to the Cauchy integral formula, and are related to integral polynomials. After studying various properties of these functions, Banach and Frechet spaces of integral holomorphic functions are defined, and several aspects investigated: duality, Taylor series approximation, biduality and reflexivity. In this paper we define and study a class of holomorphic functions over infinite- dimensional Banach spaces admitting integral representation. Our purpose, and the motivation for our definition, are two-fold: we wish to obtain an integral repre- sentation formula …

Computational aspects in 2D SBEM analysis with domain inelastic actions

2009

The Symmetric Boundary Element Method, applied to structures subjected to temperature and inelastic actions, shows singular domain integrals. In the present paper the strong singularity involved in the domain integrals of the stresses and tractions is removed, and by means of a limiting operation, this traction is evaluated on the boundary. First the weakly singular domain integral in the Somigliana Identity (S.I.) of the displacements is regularized and the singular integral is transformed into a boundary one using the Radial Integration Method; subsequently, using the differential operator applied to the displacement field, the S.I. of the tractions inside the body is obtained and through…

Solving the Cut-Off Wave Numbers in Partially filled Rectangular Waveguides with Ferrite by the Cauchy Integral Method

2005

The modal analysis of the off-centered rectangular waveguide loaded with a vertical slab of ferrite material, biased in the y-direction by a DC magnetic field, leads to the resolution of a transcendent equation whose infinite solutions are the TE/sub m0/ cutoff wave numbers in the guide. The method based on the Cauchy integral (Delvest L.M. and Lyness, J.N., 1967) is becoming very popular for solving such equations. This powerful method is described for solving the propagation constant in a partially ferrite filled waveguide. The method is used to calculate the propagation constant of the fundamental TE mode for some configurations used in the literature about ferrites. Results obtained in …

Infinite Dimensional Holomorphy

2019

We give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Frechet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous…

Holomorphic Functions on Polydiscs

2019

This is a short introduction to the theory of holomorphic functions in finitely and infinitely many variables. We begin with functions in finitely many variables, giving the definition of holomorphic function. Every such function has a monomial series expansion, where the coefficients are given by a Cauchy integral formula. Then we move to infinitely many variables, considering functions defined on B_{c0}, the open unit ball of the space of null sequences. Holomorphic functions are defined by means of Frechet differentiability. We have versions of Weierstrass and Montel theorems in this setting. Every holomorphic function on B_{c0} defines a family of coefficients through a Cauchy integral …