Isoperimetric Inequalities in Mathematical Physics. (AM-27)

Minkowski’s inequality for two variable difference means

“…. ) and by (19), we get, again by induction, the required inequality (26) are nonpositive. In T 3 ∪ T 4 we have b − 1 ≤ 0, 2a − b ≥ 0, and thus the first term of h n is nonpositive.…”

“…In T 3 we have a − 1 ≤ 0, a − 2b ≥ 0 while in T 4 the inequalities a − 1 ≥ 0, a−2b < 0 hold; hence in both cases the second term of h n is nonpositive. Using (19) we get, by induction, the inequality (25).…”

“…Some of these means have applications in electrostatics [1], [19], in heat conductions and chemical problems [26]. Of particular recent interest is the occurrence of these means in signal processing theory in connection with time-frequency distributions.…”

On the $${ L}^{ p}$$Lp norm of the torsion function

“…Iterating (36) n − 1 times we find (14). The same calculation carries over when Ω is an open, bounded convex set.…”

“…For example the torsional rigidity of a cross section of a beam appears in the computation of the angular change when a beam of a given length and a given modulus of rigidity is exposed to a twisting moment [1,14].…”