On approximation of superharmonic functions in open sets

M. Shirinbekov

Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 591-599

Résumé

This article deals with an investigation and some applications of the following problem. Let $D\subset\mathbf R^n$, $n\geqslant2$, be a bounded region coinciding with the interior of its closure, let $S(\overline D)$ be the set of bounded superharmonic functions on $D$, and let $S_C^0(\overline D)$ be the set of functions continuous and superharmonic in a neighborhood of $\overline D$. It is necessary to find conditions under which each function $V(x)$ in some subset $S'\subset S(D)$ is representable in the form $$ V(x)=\varliminf_{y\to x}\inf F(y),\qquad x,y\in D, $$ where the infimum is over a system of functions in $S_C^0(D)$ such that $F(x)>\overline V(x)=\varlimsup_{y\to x}V(y)$, $x,y\in D$. A solution is presented for certain cases when the set $S'$ is specified concretely. Bibliography: 9 titles.