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- Perron_method 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 ifdiverges.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.".
- Perron_method wikiPageExternalLink BF01180608.
- Perron_method wikiPageExternalLink item?id=ASNSP_1963_3_17_1-2_43_0.
- Perron_method wikiPageID "26444076".
- Perron_method wikiPageRevisionID "592766436".
- Perron_method author "Solomentsev, E.D.".
- Perron_method hasPhotoCollection Perron_method.
- Perron_method id "p/p072370".
- Perron_method title "Perron method".
- Perron_method subject Category:Partial_differential_equations.
- Perron_method subject Category:Potential_theory.
- Perron_method subject Category:Subharmonic_functions.
- Perron_method type Abstraction100002137.
- Perron_method type Communication100033020.
- Perron_method type DifferentialEquation106670521.
- Perron_method type Equation106669864.
- Perron_method type Function113783816.
- Perron_method type MathematicalRelation113783581.
- Perron_method type MathematicalStatement106732169.
- Perron_method type Message106598915.
- Perron_method type PartialDifferentialEquation106670866.
- Perron_method type PartialDifferentialEquations.
- Perron_method type Relation100031921.
- Perron_method type Statement106722453.
- Perron_method type SubharmonicFunctions.
- Perron_method 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.".
- Perron_method label "Perron method".
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- Perron_method sameAs Q7169664.
- Perron_method sameAs Q7169664.
- Perron_method sameAs Perron_method.
- Perron_method wasDerivedFrom Perron_method?oldid=592766436.
- Perron_method isPrimaryTopicOf Perron_method.