Speaker
Description
We consider integral inequalities for functions $f\in C_0^{\infty}(\Omega)$ on domains $\Omega$ of the Euclidean space $\mathbb{R}^n$, $n\in \mathbb{N}$.
There are several inequalities connected with two following basic results. Namely, in 1954 F. Rellich proved that
$$
\int_{\mathbb{R}^n} |\Delta f|^2 \,dx\geq
\frac{n^2(n-4)^2}{16}\int_{\mathbb{R}^n}\frac{|f|^2} {|x|^{4}} dx, \quad \quad
\forall f\in C_0^\infty(\mathbb{R}^n\setminus \{0\}) \,\, (n\geq 3),
$$
where $\Delta f$ is the Laplacian of the function $f:\mathbb{R}^n\setminus {0} \to\mathrm{C}$.
In 1961 M. Sh. Birman proved that
$$
\int_0^{\infty}|f^{(k)}(t)|^2 dt \geq \left(
\frac{(2k-1)!!}{2^k} \right)^2\int_0^{\infty}\frac{|f(t)|^2}{t^{2k}} dt, \quad \forall f \in C_0^\infty(0, \infty)\,\, (k\in \mathbb{N}).
$$
We will describe direct generalizations of the Rellich and Birman inequalities. In addition, using the polyharmonic operator defined by
$$
\Delta^{k/2}f(x):= \begin {cases}
\Delta^j f(x),&\text{if $k=2j$},\\
\nabla \Delta^{j-1} f(x),&\text{if $k=2j-1$},
\end {cases}
$$
where $j\in \mathbb{N}$, $k\in \mathbb{N}$, $f\in C_0^{\infty}(\Omega)$, $\Omega\subset\mathbb{R}^n$, we describe results about the inequality
$$
\int_{\Omega} |\Delta^{k/2} f(x)|^2 dx\geq
A_k(\Omega)\int_{\Omega}\frac{|f(x)|^2}{{\rm dist}^{2k}(x, \partial\Omega)}dx, \quad \forall f\in C_0^\infty(\Omega).
$$
