Reduced function

Remarks

The function $u$ itself is admissible, so $0 \le \widehat{R}^E_u \le R^E_u \le u$, and $R^E_u = u$ on $E$. By the fundamental convergence theorem, $\widehat{R}^E_u$ is superharmonic and differs from $R^E_u$ only on a polar set contained in $E$. The function $\widehat{R}^E_u$ is harmonic on $\Omega \setminus \overline{E}$. Example: let $\Omega = \R^3$, $u = 1$ and $E = \set{x : \abs{x} \le 1}$. The function $v(x) = \min(1, 1/\abs{x})$ is positive, superharmonic and equal to $1$ on $E$. Any admissible $v$ satisfies $v \ge 1/\abs{x}$ on $\abs{x} > 1$ by the minimum principle, so $R^E_1 = \widehat{R}^E_1 = \min(1, 1/\abs{x})$. Non-example of an admissible function: in the same example, the indicator function of the closed ball $E$ equals $u = 1$ on $E$. It is not superharmonic, since it is not lower semicontinuous at the points of the unit sphere. Other notations are $R_u^E$, $\operatorname{red}_E u$, and the French name réduite.