Partial differential equations I — elliptic equations

1986 ◽  
pp. 210-242
Author(s):  
Graham Vahl Davis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Partial Differential

Symmetry of solutions to optimization problems related to partial differential equations

2006 ◽  
Vol 136 (5) ◽  
pp. 921-934 ◽  
Author(s):  
Fabrizio Cuccu ◽  
Giovanni Porru
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Optimization Problems ◽  
Existence Results ◽  
Dirichlet Problems ◽  
Partial Differential ◽  
Moving Plane Method ◽  
Plane Method ◽  
Moving Plane

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

Limitations due to Noise, Stability and Component Tolerance on the Solution of Partial Differential Equations by Differential Analysers

SIMULATION ◽  
1964 ◽  
Vol 3 (5) ◽  
pp. 45-52 ◽  
Author(s):  
Michael E. Fisher
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Truncation Error ◽  
High Accuracy ◽  
Quantitative Investigation ◽  
Parallel Method ◽  
Partial Differential ◽  
Truncation Errors ◽  
The Difference

The solution of partial differential equations by a differential analyser is considered with regard to the effects of noise, computational instability and the deviation of components from their ideal values. It is shown that the 'serial' method of solving parabolic, hyperbolic and elliptic equations leads to serious in stability which increases as the finite difference interval is reduced. The truncation error (due to the difference approximations) decreases as the interval is made smaller and consequently an 'optimal' ac curacy is reached when the unstable noise errors match the truncation errors. Evaluation shows that the attainable accuracy is severely limited, especially for hyperbolic and elliptic equations. The 'parallel' method is stable when applied to parabolic and hyperbolic (but not elliptic) equations and the attainable accuracy is then limited by the accumulation of component tolerances. Quantitative investigation shows how reasonably high accuracy can be achieved with a minimum of precise adjustments.

Normal solutions of elliptic equations

1996 ◽  
Vol 119 (2) ◽  
pp. 363-371 ◽  
Author(s):  
Pekka Koskela
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Harmonic Functions ◽  
Elliptic Partial Differential Equations ◽  
Boundary Behaviour ◽  
Partial Differential ◽  
Sobolev Functions ◽  
Monotone Sobolev Functions

AbstractWe extend a number of known criteria for normality of analytic and harmonic functions to the setting of solutions to elliptic partial differential equations. Some of the results hold for monotone Sobolev functions. We also discuss the boundary behaviour of monotone Sobolev functions.

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