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In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. 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. where is the set of all subharmonic functions such that on the boundary of the domain. diverges.

Attributes	Values
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rdfs:label	Perron method (en)
rdfs:comment	In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. 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. where is the set of all subharmonic functions such that on the boundary of the domain. diverges. (en)
dcterms:subject	Potential theory Partial differential equations Subharmonic functions
Wikipage page ID	26444076 (xsd:integer)
Wikipage revision ID	1117739042 (xsd:integer)
Link from a Wikipage to another Wikipage	Potential theory Mathematische Zeitschrift Elliptic operator Subharmonic function Norbert Wiener Oskar Perron Dirichlet problem Graduate Texts in Mathematics Hans Weinberger Guido Stampacchia Potential theory Partial differential equations Laplace's equation Dover Publications Subharmonic functions Capacity of a set Springer-Verlag Harmonic functions
Link from a Wikipage to an external page	http://www.numdam.org/item%3Fid=ASNSP_1963_3_17_1-2_43_0
sameAs	Perron method Perron method Perron method Perron method
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Short_description dbt:MathSciNet dbt:Mathanalysis-stub
author	Solomentsev, E.D. (en)
id	p/p072370 (en)
title	Perron method (en)
has abstract	In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. 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 Dirichlet problem is to find a harmonic function in a domain, with boundary conditions given by a continuous function . The Perron solution is defined by taking the pointwise supremum over a family of functions , where is the set of all subharmonic functions such that on the boundary of the domain. The Perron solution u(x) is always harmonic; however, the values it takes on the boundary may not be the same as the desired boundary values . A point y of the boundary satisfies a barrier condition if there exists a superharmonic function , defined on the entire domain, such that and for all . Points satisfying the barrier condition are called regular points of the boundary for the Laplacian. These are precisely the points at which one is guaranteed to obtain the desired boundary values: as . The characterization of regular points on surfaces is part of potential theory. Regular points on the boundary of a domain are those points that satisfy the Wiener criterion: for any , let be the capacity of the set ; then is a regular point if and only if diverges. The Wiener criterion was first devised by Norbert Wiener; it was extended by Werner Püschel to uniformly elliptic divergence-form equations with smooth coefficients, and thence to uniformly elliptic divergence form equations with bounded measureable coefficients by Walter Littman, Guido Stampacchia, and Hans Weinberger. (en)
prov:wasDerivedFrom	wikipedia-en:Perron_method?oldid=1117739042&ns=0
page length (characters) of wiki page	4906 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Perron_method
is Link from a Wikipage to another Wikipage of	Perron's method Oskar Perron Dirichlet problem Uniformization theorem List of things named after Norbert Wiener Viscosity solution
is Wikipage redirect of	Perron's method
is known for of	Oskar Perron
is known for of	Oskar Perron
is foaf:primaryTopic of	wikipedia-en:Perron_method

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