Perron's method: Difference between revisions
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imported>RayAYang (brief stub, fill out more later when I can get home to my references.) |
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</ref> The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble. | </ref> The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble. | ||
The Perron method can be used [[viscosity solutions]] whenever a [[comparison principle]] is available | The Perron method can be used [[viscosity solutions]] whenever a [[comparison principle]] is available and appropriate barriers can be constructed to assure the boundary conditions. The Perron solution is taken to be the largest viscosity subsolution, or the least viscosity supersolution. | ||
==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
Latest revision as of 00:28, 29 January 2012
Perron's method, also known as the method of subharmonic functions, is a technique originally introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation.[1] The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble.
The Perron method can be used viscosity solutions whenever a comparison principle is available and appropriate barriers can be constructed to assure the boundary conditions. The Perron solution is taken to be the largest viscosity subsolution, or the least viscosity supersolution.
References
- ↑ Perron, O. (12 1923), "Eine neue Behandlung der ersten Randwertaufgabe für Δu=0", Mathematische Zeitschrift (Springer Berlin / Heidelberg) 18 (1): 42–54, doi:10.1007/BF01192395, ISSN 0025-5874