Hardy-type inequalities
Though the well-known Hardy-Sobolev inequality \[ K^2\int_\Omega u^2/| x|^2\,dx\leq\int_\Omega|\nabla u|^2\,dx, \quad \Omega\subset\mathbb R^n,\;n\geq3,\;u\in C^\infty_c(\Omega), \] holds with an optimal constant, the equality is never achieved. Therefore the inequality has been improved in various ways. Recall, for instance, the inequality \[ \gamma \biggl(\int_\Omega| u|^p\biggr)^{2/p}+ K^2\int_\Omega u^2/| x|^2 \leq \int_\Omega |\nabla u|^2, \quad \gamma>0, \;\Omega \text{ bounded, }1\leq p<2n/(n-2), \] proved by \textit{H. Brézis} and \textit{J. L. Vázquez} [Rev. Mat. Univ. Complutense Madr. 10, No. 2, 443--469 (1997; Zbl 0894.35038)]. The authors consider the operators of the form \(L=-\Delta-\mu/d(x)^2\), where \(\mu\in\mathbb R^n\), \(d(x)=\text{dist}(x,\Sigma)\), \(x\in\Omega\), \(\Omega\subset\mathbb R^n\) is an open bounded set, \(\Sigma\) is a compact smooth manifold without boundary of codimension \(k\neq2\), and in this connection they are interested in the question whether an inequality analogous to the above one holds. They prove that there exist constants \(C>0\), \(\gamma>0\) such that any \(u\in C^\infty_c(\Omega\setminus\Sigma)\) satisfies \[ \gamma\biggl(\int_\Omega| u|^p\biggr)^{2/p}+ H^2\int_\Omega u^2/d^2 \leq \int_\Omega |\nabla u|^2+ C\int_\Omega u^2, \] where \(d(x)=\text{dist}(x,\Sigma)\), \(H=(k-2)/2\), \(1\leq p<p_k\) and \(p_k\) is given by \(\frac1{p_k}=\frac12-\frac2{k(n-k+2)}\) for \(k>2\), \(\frac1{p_1}=\frac12-\frac1{n+1}\) for \(k=1\). Moreover, there exist \(\beta>0\) and a neighbourhood \(\Omega_\beta=\{x\in\Omega: d(x,\Sigma)<\beta\}\) of \(\Sigma\) such that for any \(u\in C^\infty_c(\Omega_\beta\subset\Sigma)\) the inequality \[ H^2\int_\Omega u^2/d^2\leq\int_\Omega|\nabla u|^2 \] holds. The authors admit that the values of \(p_k\) are probably not optimal and they expect that these values might be replaced with \(2n/(n-2)\). Finally, they extend some of their earlier results [Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 1, 61--83 (2003; Zbl 1040.35006)] deriving pointwise estimates for solutions of the linear boundary value problem \(-\Delta u-a(x)u=f\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), under the assumptions that \(a\in L^1_{\text{loc}}(\Omega)\), \(a\) is bounded below and \[ \gamma\biggl(\int_\Omega| u|^r\biggr)^{2/r}+ \int_\Omega a(x)u^2 \leq \int_\Omega|\nabla u|^2 + M\int_\Omega u^2 \] for some \(r>2\), \(\gamma>0\), \(M>0\).
scientific article; zbMATH DE number 2102879