Some multidimensional opial type inequalities

In this note, we establish some multidimensional Opial type inequalities, which are generalized from the one-dimensional case. Based on calculus and some fundamental inequalities, we first present an elementary proof in Euclidean spaces. Then by using the property of the Minkowski gradient and the so called adapted frame field, we further extend the multidimensional Opial type inequalities to a Minkowski space.