# 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  gave the initial presentation of the method, which was substantially developed by N. Wiener

and M.V. Keldysh .

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 , where an example is also constructed of a regular domain $\Omega$ within which the Dirichlet problem is unstable).