A note on multidimensional integral inequalities (Q1093753)

Let B be a bounded domain in \({\mathbb{R}}^ n\), \(n\geq 2\), and \(u_ 1,...,u_ m\) be real-valued sufficiently smooth functions which vanish on the boundary of B. The author used the same elementary methods as e.g. in his paper in J. Math. Anal. Appl. 114, 111-115 (1986; Zbl 0599.26028) in order to prove inequalities between geometric and arithmetic means of \(L^ 1\)- or \(L^ 2\)-norms of \(u_ i\) and/or \(\text{grad} u_ i\) and/or \(\Delta u_ i,\) \(i=1,...,m.\)