scholarly journals Existence and uniqueness of global solutions to fully nonlinear first order elliptic systems

2015 ◽  
Vol 115 ◽  
pp. 50-61 ◽  
Author(s):  
Nikos Katzourakis
Keyword(s):  
Existence And Uniqueness ◽  
Global Solutions ◽  
Elliptic Systems ◽  
Fully Nonlinear ◽  
First Order

scholarly journals On the well-posedness of global fully nonlinear first order elliptic systems

2018 ◽  
Vol 7 (2) ◽  
pp. 139-148 ◽  
Author(s):  
Hussien Abugirda ◽  
Nikos Katzourakis
Keyword(s):  
Elliptic Systems ◽  
Order System ◽  
Well Posedness ◽  
Unique Strong Solution ◽  
Ellipticity Condition ◽  
Fully Nonlinear ◽  
First Order ◽  
Classical Work ◽  
Order Case ◽  
Well Posed

AbstractIn the very recent paper [15], the second author proved that for any {f\in L^{2}(\mathbb{R}^{n},\mathbb{R}^{N})}, the fully nonlinear first order system {F(\,\cdot\,,\mathrm{D}u)=f} is well posed in the so-called J. L. Lions space and, moreover, the unique strong solution {u\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}} to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for F inspired by Campanato’s classical work in the 2nd order case. Herein, we extend the results of [15] by introducing a new strictly weaker ellipticity condition and by proving well-posedness in the same “energy” space.

Schwarz problem for first-order elliptic systems on the plane

2017 ◽  
Vol 53 (10) ◽  
pp. 1318-1328 ◽  
Author(s):  
V. B. Vasil’ev ◽  
V. G. Nikolaev
Keyword(s):  
Elliptic Systems ◽  
First Order ◽  
Schwarz Problem

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

ON THE FIRST BOUNDARY VALUE PROBLEM FOR THE CLASS OF ELLIPTIC SYSTEMS DEGENERATING AT AN INNER POINT

2001 ◽  
Vol 6 (1) ◽  
pp. 147-155 ◽  
Author(s):  
S. Rutkauskas
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Second Order ◽  
Type Problem ◽  
Dirichlet Type ◽  
Holder Class ◽  
Uniqueness Of The Solution ◽  
Dirichlet Type Problem

The Dirichlet type problem for the weakly related elliptic systems of the second order degenerating at an inner point is discussed. Existence and uniqueness of the solution in the Holder class of the vector‐functions is proved.

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