5. An integral inequality

The following paper presents an integral inequality of a nonclassical kind, motivated by an extremum problem in continuum mechanics:

• An Integral Inequality and Plastic Torsion, Archive for Rational Mechanics and Analysis 72:1(1979), 23-39.

The inequality in its simplest form reads:

Let $\Omega \subset R^2$ be a bounded domain. Let $f(x,y)$ be a Lipschitz function on $\overline {\Omega}$ and $f=0$ on $\partial\Omega$. Put $$ F(\theta)= m\{x:x\in\Omega, |grad\, f(x)|\le \theta \}, $$ and $M =|grad\,f|_{L^\infty}$. Then $$ \biggl|\int\!\!\!\int_{\Omega} f\,dxdy\biggr| \le \frac{1}{3\sqrt{\pi}}\int_0^M ((m\Omega)^{3/2}-F(\theta)^{3/2})d\theta. $$

Equality holds if and only if\, $\Omega$ is an open circle and $f$ or $-f$ is a dome function.
(A dome function is a concave, non-increasing function of the distance to some fixed point.)
The inequality is extended to functions in the Sobolev space $W^{1,1}(R^n)$, having a support of finite measure, in the paper

• Estimating the Integral of a Function in Terms of a Distribution Function of Its Gradient, by G.A. and G. Talenti, Bollettino U. M. I. 18:B(1981), 885-894.

The condition for equality is completely analogous to the condition for the first case.

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Last updated: 2019-11-29