Isoperimetric and functional inequalities

We establish lower estimates for an integral functional
$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$
where $\Omega$ — a bounded domain in $\mathbb{R}^n \; (n \geqslant 2)$, an integrand $f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)$ – a function that is $B$-measurable with respect to a variable $t$ and is convex and even in the variable $p$, $\nabla u(x)$ — a gradient (in the sense of Sobolev) of the function $u \colon \Omega \rightarrow \mathbb{R}$. In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality $H^{n-1}( \partial A) \geqslant \lambda(m_n A)$, that connects $(n-1)$-dimensional Hausdorff measure $H^{n-1}(\partial A )$ of relative boundary $\partial A$ of the set $A \subset \Omega$ with its $n$-dimensional Lebesgue measure $m_n A$. The integrand $f$ is assumed to be isotropic, i.e. $f(t,p) = f(t,q)$ if $|p| = |q|$. Applications of the established results to multidimensional variational problems are outlined. For functions $ u $ that vanish on the boundary of the domain $\Omega$, the assumption of the isotropy of the integrand $ f $ can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand $ f $ and of the function $ u $. The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function $u$, but also to the integrand $f$. The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.