Poisson modification on a compactly contained disc

Let $u$ be subharmonic on a complex domain $\mathrm{\Omega }$, and let $D=D(a,r)⋐\mathrm{\Omega }$ be an open disc. A boundary approximation for $u$ on $\mathrm{\partial }D$ is a decreasing sequence of continuous functions ${\varphi }_n:\mathrm{\partial }D\to \mathbb{R}$ with ${\varphi }_n\downarrow u{\mathrm{\mid }}_{\mathrm{\partial }D}$; such sequences exist because the circle data are upper semicontinuous by Upper semicontinuous functions are Borel and their circle averages are defined.

For each $n$, let $h_n$ be the harmonic function on $D$, continuous on $\bar{D}$, with boundary values ${\varphi }_n$, obtained by transporting the unit-disc Poisson solution of The Poisson integral gives the unique continuous harmonic extension on the closed unit disc across the affine map $z\mapsto (z-a)\mathrm{/}r$ and using Plane harmonicity is preserved by holomorphic and antiholomorphic changes of coordinate.

The Poisson modification of $u$ on $D$ is the function $P_{D}u$ defined by $P_{D}u(z)=\{\begin{array}{ll}{\inf }_n{h}_n(z),& z\in D,\\ u(z),& z\in \mathrm{\Omega }\setminus D\mathrm{.}\end{array}$

The next theorem proves that the inside function ${\inf }_n{h}_n$ is harmonic, independent of the chosen boundary approximation, and no smaller than the original subharmonic function on $D$.