scholarly journals The Method of Fischer-Riesz Equations for Elliptic Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
A. Alsaedy ◽  
N. Tarkhanov
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Cauchy Data ◽  
Adjoint Problem ◽  
Green Formula ◽  
Order System ◽  
Elliptic Boundary Value ◽  
The Given ◽  
The Green Formula

We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first-order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well-elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.

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.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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