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Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of ℂ ∞ ∖ E with pole at infinity and M ₙ ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E ≢ 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z ∈ E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ₙ ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

The Functional Equation

Pavlos Sinopoulos (1987)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

The successive approximation method for the Dirichlet problem in a planar domain

Dagmar Medková (2008)

Applicationes Mathematicae

The Dirichlet problem for the Laplace equation for a planar domain with piecewise-smooth boundary is studied using the indirect integral equation method. The domain is bounded or unbounded. It is not supposed that the boundary is connected. The boundary conditions are continuous or p-integrable functions. It is proved that a solution of the corresponding integral equation can be obtained using the successive approximation method.

Theoretical and numerical study of a free boundary problem by boundary integral methods

Michel Crouzeix, Philippe Féat, Francisco-Javier Sayas (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.

Theoretical and numerical study of a free boundary problem by boundary integral methods

Michel Crouzeix, Philippe Féat, Francisco-Javier Sayas (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.

Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition

Mario Durán, Eduardo Godoy, Jean-Claude Nédélec (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green's function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Durán et al. (cf. [Numer. Math.107 (2007) 295–314; IMA J. Appl. Math.71 (2006) 853–876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important...

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