In the mathematical study of
harmonic functions
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
, the Perron method, also known as the method of
subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of function (mathematics), functions used extensively in partial differential equations, complex analysis and potential theory.
Intuitively, subharmonic functions are re ...
s, is a technique introduced by
Oskar Perron for the solution of the
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
for
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
. 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
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
. 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
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
; it was extended by Werner Püschel to uniformly
elliptic
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
divergence-form equations with smooth coefficients, and thence to uniformly elliptic divergence form equations with bounded measureable coefficients by Walter Littman,
Guido Stampacchia
Guido Stampacchia (26 March 1922 – 27 April 1978) was an Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations..
Life and acade ...
, and
Hans Weinberger.
References
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Further reading
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Potential theory
Partial differential equations
Subharmonic functions
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