# 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).

#### 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