Perron method

A method for solving the Dirichlet problem for the Laplace equation based on the properties of subharmonic functions (and superharmonic functions, cf. Subharmonic function). O. Perron [1] gave the initial presentation of the method, which was substantially developed by N. Wiener

and M.V. Keldysh [4].

Let $ \Omega $ be a bounded domain in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, with boundary $ \Gamma = \partial \Omega $, let $ f= f( y) $ be a real-valued function on $ \Gamma $, $ - \infty \leq f( y) \leq + \infty $. Let $ \Phi $ be the non-empty family of all superharmonic functions $ v( x) $, $ x \in \Omega $, in the wide sense (i.e. the function $ v( x) \equiv + \infty $ belongs to $ \Phi $) that are bounded from below and are such that

$$ \lim\limits _ {x \rightarrow y } \inf v( x) \geq f( y),\ \ y \in \Gamma . $$

Let

$$ \overline{H}\; _ {f} ( x) = \overline{H}\; _ {f} ( x; \Omega ) = \inf \{ {v( x) } : {v \in \Phi } \} ,\ \ x \in \Omega , $$

be the lower envelope of $ \Phi $. Along with $ \Phi $, consider the non-empty family $ \Psi $ of all subharmonic functions $ u( x) $, $ x \in \Omega $, in the wide sense (the function $ u( x) \equiv - \infty \in \Psi $) that are bounded from above and are such that

$$ \lim\limits _ {x \rightarrow y } \sup u( x) \leq f( y) ,\ \ y \in \Gamma . $$

Let

$$ \underline{H} {} _ {f} ( x) = \underline{H} {} _ {f} ( x; \Omega ) = \sup \{ {u( x) } : {u \in \Psi } \} ,\ \ x \in \Omega , $$

be the upper envelope of $ \Psi $.

There are only three possibilities for $ \overline{H}\; _ {f} $( and $ \underline{H} {} _ {f} $): $ \overline{H}\; _ {f} ( x) \equiv + \infty $, $ \overline{H}\; _ {f} ( x) \equiv - \infty $ or $ \overline{H}\; _ {f} ( x) $ is a harmonic function; and always

$$ \underline{H} {} _ {f} ( x) \leq \overline{H}\; _ {f} ( x) ,\ \ x \in \Omega . $$

The function $ f( y) $, $ y \in \Gamma $, is called resolutive if the two envelopes $ \overline{H}\; _ {f} $ and $ \underline{H} {} _ {f} $ are finite and coincide. In that case the harmonic function $ H _ {f} = \overline{H}\; _ {f} = \underline{H} {} _ {f} $ is the generalized solution to the Dirichlet problem for the function $ f( y) $, $ y \in \Gamma $( in the sense of Wiener–Perron). For $ f( y) $, $ y \in \Gamma $, to be resolutive it is necessary and sufficient that it be integrable with respect to the harmonic measure on $ \Gamma $( Brelot's theorem). Any continuous finite function $ f( y) $, $ y \in \Gamma $, is resolutive (Wiener's theorem).

A point $ y _ {0} \in \Gamma $ is called a regular boundary point if the following limit relation applies for any continuous finite function $ f( y) $, $ y \in \Gamma $:

$$ \lim\limits _ {x \rightarrow y _ {0} } H _ {f} ( x) = f( y _ {0} ). $$

Regularity at all points $ y \in \Gamma $ is equivalent to the existence of classical solutions $ w _ {f} ( x) $ to the Dirichlet problem for any continuous finite function $ f( y) $, $ y \in \Gamma $, and in that case $ H _ {f} ( x) \equiv w _ {f} ( x) $; a bounded domain $ \Omega $ all boundary points of which are regular is sometimes also called regular. For a point $ y _ {0} \in \Gamma $ to be regular it is necessary and sufficient that there is a barrier at $ y _ {0} $.

Points $ y _ {0} \in \Gamma $ that are not regular are called irregular boundary points. For example, isolated points are irregular boundary points, as are the vertices of sufficiently sharp wedges entering $ \Omega $ if $ n \geq 3 $( Lebesgue spines). The set of all irregular points of $ \Gamma $ is a set of type $ F _ \sigma $ of capacity zero.

Let there be a sequence of domains $ \Omega _ {k} $, $ \overline \Omega \; _ {k} \subset \Omega _ {k+} 1 $, such that $ \Omega = \cup _ {k=} 1 ^ \infty \Omega _ {k} $, and let a continuous finite function $ f( y) $, $ y \in \Gamma $, be continuously extendible to $ \Gamma $. Then

$$ \lim\limits _ {k \rightarrow \infty } H _ {f} ( x; \Omega _ {k} ) = H _ {f} ( x; \Omega ),\ \ x \in \Omega , $$

uniformly on compact sets in $ \Omega $; in the case of regular domains $ \Omega _ {k} $ one obtains a construction à la Wiener for the generalized solution to the Dirichlet problem. Now consider an arbitrary sequence of domains $ G _ {k} $, $ \partial G _ {k} \rightarrow \Gamma $, $ G _ {k} \supset \overline \Omega \; $, for a domain $ \Omega $ without an interior boundary. In that case, in general

$$ \lim\limits _ {k \rightarrow \infty } H _ {f} ( x; G _ {k} ) \neq H _ {f} ( x; \Omega ). $$

The Dirichlet problem is stable in a domain $ \Omega $ or in a closed domain $ \overline \Omega \; $ if

$$ \lim\limits _ {k \rightarrow \infty } H _ {f} ( x; G _ {k} ) = H _ {f} ( x; \Omega ) $$

for all $ x \in \Omega $ or for all $ x \in \overline \Omega \; $, respectively. For the Dirichlet problem to be stable in a domain $ \Omega $ it is necessary and sufficient that the sets of all irregular points in the complements $ C \Omega $ and $ C \overline \Omega \; $ coincide; stability in a closed domain requires that $ C \overline \Omega \; $ does not have irregular points (Keldysh' theorems, cf. Keldysh theorem and [4], where an example is also constructed of a regular domain $ \Omega $ within which the Dirichlet problem is unstable).

See also Upper-and-lower-functions method.

References

[1]	O. Perron, "Eine neue Behandlung der ersten Randwertaufgabe für " Math. Z. , 18 (1923) pp. 42–54 MR1544619 Zbl 49.0340.01
[2]	I.G. Petrovskii, "Perron's method for the solution of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 107–114 (In Russian)
[3a]	N. Wiener, "Certain notions in potential theory" J. Math. Phys. , 3 (1924) pp. 24–51 Zbl 50.0646.03 Zbl 51.0360.05
[3b]	N. Wiener, "The Dirichlet problem" J. Math. Phys. , 3 (1924) pp. 127–146 MR1500496 Zbl 50.0646.02 Zbl 50.0335.04 Zbl 50.0335.02 Zbl 51.0361.01
[3c]	N. Wiener, "Note on paper of O. Perron" J. Math. Phys. , 4 (1925) pp. 21–32 Zbl 51.0365.06 Zbl 51.0361.02
[4]	M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–231 (In Russian) Zbl 0179.43901
[5]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903

Comments

Counterexamples from S. Zaremba [a5] and H. Lebesgue [a2] made it clear that the existence of a classical solution to the Dirichlet problem for the Laplace equation is not always guaranteed. Lebesgue [a3] therefore proposed to construct a solution operator from the space of boundary functions into the set of harmonic functions on the domain. This operator should be linear and isotone (cf. Isotone mapping), and produce the classical solution if one exists. N. Wiener [3c] showed that such a solution operator for continuous boundary functions is obtained from the Perron method. The method is extended to arbitrary boundary functions by M. Brelot, and, since then, is called the Perron–Wiener–Brelot method. The uniqueness of the solution operator is proved by M.V. Keldysh (cf. Keldysh theorem). All these results have their counterpart in the abstract theory of harmonic spaces (cf. Harmonic space), cf. [a4].

References

[a1]	M. Brelot, "Familles de Perron et problème de Dirichlet" Acta Sci. Math. (Szeged) , 9 (1938–1940) pp. 133–153 MR0000734 Zbl 0023.23302 Zbl 0021.13102 Zbl 65.0419.01 Zbl 65.0418.03
[a2]	H. Lebesgue, "Sur des cas d'impossibilité du problème de Dirichlet ordinaire" C.R. Séances Soc. Math. France , 41 (1913) pp. 17
[a3]	H. Lebesgue, "Conditions de régularité, conditions d'irrégularité, conditions d'impossibilité dans le problème de Dirichlet" C.R. Acad. Sci. , 178 (1924) pp. 349–354
[a4]	I. Netuka, "The Dirichlet problem for harmonic functions" Amer. Math. Monthly , 87 (1980) pp. 621–628 MR0600920 Zbl 0454.31002
[a5]	S. Zaremba, "Sur le principe de Dirichlet" Acta Math. , 34 (1911) pp. 293–316 MR1555069 Zbl 42.0393.01