Separately harmonic and subharmonic functions
Let \(u(x,y)\) be defined on \(B_1 \times B_2\), where \(B_1\) is the unit ball of \(\mathbb{R}^m\) and \(B_2\) is the unit ball of \(\mathbb{R}^n\). It is known that, if \(u(x, \cdot)\) is harmonic on \(B_2\) for each \(x\) and \(u (\cdot,y)\) is harmonic on \(B_1\) for each \(y\), then \(u\) is harmonic on \(B_1 \times B_2\). This statement fails if ``harmonic'' is replaced by ``subharmonic'' throughout. A hybrid problem, which has remained unsolved for some time, concerns the case where \(u(x, \cdot)\) is harmonic on \(B_2\) for each \(x\) and \(u(\cdot,y)\) is subharmonic on \(B\) for each \(y\). Is \(u\) then subharmonic on \(B_1 \times B_2\)? \textit{S. A. Imomkulov} [Dokl. Akad. Nauk UzSSR 1990, No. 2, 8-10 (1990; Zbl 0708.31005)], has pointed out that the answer is ``yes'' under the rather strong additional assumption that \(u(\cdot,y)\) is real analytic for each \(y\). The present paper shows, by a different argument, that it is enough to assume that \(u(\cdot,y)\) is of class \(C^2\) for each \(y\). The general problem of whether one needs any assumption on \(u(\cdot,y)\), apart from subharmonicity, remains open.
scientific article; zbMATH DE number 936755
scientific article