On BVPS for Strongly Elliptic Systems with Higher Order Boundary Conditions

2007 ◽  
Vol 14 (1) ◽  
pp. 145-167
Author(s):  
Flavia Lanzara
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Singular Integral ◽  
Elliptic Systems ◽  
Higher Order ◽  
Boundary Operator ◽  
Compatibility Conditions ◽  
Layer Potential ◽  
Integral System ◽  
Regular Type

Abstract We consider BVPs for strongly elliptic systems of order 2𝑙 with the boundary conditions of order 𝑙 + 𝑛, 𝑛 ⩾ 0. By representing the solution by means of a simple layer potential of order 𝑛 and by imposing the boundary conditions, we get a singular integral system which is of regular type if and only if the boundary operator satisfies the Lopatinskiĭ condition and which can be solved if suitable compatibility conditions are satisfied. An explicit formula for computing the index of the BVP is given.

First-order elliptic systems degenerated at the boundary

1993 ◽  
Vol 123 (6) ◽  
pp. 1203-1212
Author(s):  
Abduhamid Dzhuraev
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Elliptic Systems ◽  
Bounded Solutions ◽  
Order Linear ◽  
Multiply Connected ◽  
First Order

SynopsisIn this paper we state that the bounded solutions of general first order linear elliptic systems of two equations in bounded multiply-connected plane domains, degenerated at the boundary, are determined in domains without any boundary conditions, provided the boundary is not characteristic for this system. The explicit formula for calculating the index of the system is derived.

scholarly journals Liouville-type theorem for higher-order Hardy-Hénon system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kui Li ◽  
Zhitao Zhang
Keyword(s):  
Elliptic System ◽  
Type Theorem ◽  
Elliptic Systems ◽  
Higher Order ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Integral System ◽  
Moving Planes ◽  
Polyharmonic System ◽  
Moving Spheres

<p style='text-indent:20px;'>In this paper, we study higher-order Hardy-Hénon elliptic systems with weights. We first prove a new theorem on regularities of the positive solutions at the origin, then study equivalence between the higher-order Hardy-Hénon elliptic system and a proper integral system, and we obtain a new and interesting Liouville-type theorem by methods of moving planes and moving spheres for integral system. We also use this Liouville-type theorem to prove the Hénon-Lane-Emden conjecture for polyharmonic system under some conditions.</p>

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková
Keyword(s):  
Stress Distribution ◽  
Fracture Mechanics ◽  
Boundary Conditions ◽  
Displacement Field ◽  
Singular Term ◽  
Crack Tip ◽  
Higher Order ◽  
Precise Description ◽  
Accurate Knowledge ◽  
Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

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