On Approximation of Solutions of Elliptic Boundary Value Problems for Petrovskii Elliptic Systems by Linear Combinations of Fundamental Solutions

1995 ◽  
pp. 285-290
Author(s):  
Inna Roitberg
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Fundamental Solutions ◽  
Elliptic Boundary Value Problems ◽  
Linear Combinations ◽  
Elliptic Boundary Value

scholarly journals The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

2006 ◽  
Vol 9 ◽  
pp. 287-329 ◽  
Author(s):  
Katsiaryna Krupchyk ◽  
Jukka Tuomela
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Formal Theory ◽  
Elliptic Boundary Value Problems ◽  
Constructive Method ◽  
Elliptic Boundary Value ◽  
The Given ◽  
Well Posed

AbstractElliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

scholarly journals Fundamental solutions and surface distributions

1969 ◽  
Vol 16 (3) ◽  
pp. 255-257
Author(s):  
R. A. Adams ◽  
G. F. Roach
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problems ◽  
Bounded Domain ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Second Order ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Image Position

When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formulawhere u(x)∈C2(D)∩C′(D̄) is a solution of the second order self-adjoint elliptic equationand denotes differentiation along the inward normal to ∂D at x∈∂D.

scholarly journals Singular Integral Operators and Elliptic Boundary-Value Problems. I

2017 ◽  
Vol 63 (1) ◽  
pp. 1-189
Author(s):  
Alexandre P Soldatov
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Integral Operators ◽  
Elliptic Systems ◽  
Elliptic Boundary Value Problems ◽  
Main Role ◽  
Cauchy Type ◽  
Elliptic Boundary Value ◽  
First Order

The book consists of three Parts I-III and Part I is presented here. In this book, we develop a new approach mainly based on the author’s papers. Many results are published here for the first time. Chapter 1 is introductory. The necessary background from functional analysis is given there for completeness. In this book, we mostly use weighted Ho¨lder spaces, and they are considered in Ch. 2. Chapter 3 plays the main role: in weighted Ho¨lder spaces we consider there estimates of integral operators with homogeneous difference kernels, which cover potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Ch. 4, analogous estimates are established in weighted Lebesgue spaces. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrixes. Investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

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