A Priori Inequalities and Pointwise Bounds for Solutions of Fourth Order Elliptic Partial Differential Equations

1967 ◽  
Vol 15 (5) ◽  
pp. 1136-1155 ◽  
Author(s):  
V. G. Sigillito
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
A Priori ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential

Higher-order estimates for fully nonlinear difference equations. I

2000 ◽  
Vol 43 (3) ◽  
pp. 485-510 ◽  
Author(s):  
Derek W. Holtby
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Equations ◽  
A Priori ◽  
Elliptic Partial Differential Equations ◽  
Uniform Mesh ◽  
Partial Differential ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Special Case

AbstractThe purpose of this work is to establish a priori C2, α estimates for mesh function solutions of nonlinear positive difference equations in fully nonlinear form on a uniform mesh, where the fully nonlinear finite-difference operator ℱh is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We deal here with the special case that the operator does not depend explicitly upon the independent variables. We do this by discretizing the approach of Evans for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger. The result in this special case forms the basis for a more general result in part II. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior C2, α semi-norm in terms of the C0 norm.

scholarly journals Existence of Positive Solution for a Nonlinear Weighted Bi-Harmonic System of Elliptic Partial Differential Equations via Fixed-Point Argument

2020 ◽  
Vol 40 (1) ◽  
pp. 54-70
Author(s):  
Md Asaduzzamana ◽  
Md Zulfikar Ali
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Positive Solution ◽  
A Priori ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Existence Criterion ◽  
Point Argument ◽  
Harmonic System

In this paper, we establish an existence criterion of positive solution for a nonlinear weighted bi-harmonic system of elliptic partial differential equations in the unit ball in Nn ( dimensionaleuclideanspace) The analysis of this paper is based on a topological method (a fixed-point argument). Initially, we establish a priori solution estimates, and then use a fixed-point theorem for deducing the existence of positive solutions. Finally, we prove a non-existence criterion as the complement of existence criterion. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 54-70

A New Fourth-Order Compact Off-Step Discretization for the System of 2D Nonlinear Elliptic Partial Differential Equations

2012 ◽  
Vol 2 (1) ◽  
pp. 59-82 ◽  
Author(s):  
R. K. Mohanty ◽  
Nikita Setia
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Elliptic Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Polar Coordinates ◽  
Partial Differential ◽  
Nonlinear Elliptic ◽  
Physical Problems ◽  
Grid Points

AbstractThis paper discusses a new fourth-order compact off-step discretization for the solution of a system of two-dimensional nonlinear elliptic partial differential equations subject to Dirichlet boundary conditions. New methods to obtain the fourth-order accurate numerical solution of the first order normal derivatives of the solution are also derived. In all cases, we use only nine grid points to compute the solution. The proposed methods are directly applicable to singular problems and problems in polar coordinates, which is a main attraction. The convergence analysis of the derived method is discussed in detail. Several physical problems are solved to demonstrate the usefulness of the proposed methods.

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