On the Variational Constant Associated to the -Hardy Inequality

Abstract

Let$\unicode[STIX]{x1D6FA}$be a domain in$\mathbb{R}^{m}$with nonempty boundary. In Ward [‘On essential self-adjointness, confining potentials and the$L_{p}$-Hardy inequality’, PhD Thesis, NZIAS Massey University, New Zealand, 2014] and [‘The essential self-adjointness of Schrödinger operators on domains with non-empty boundary’,Manuscripta Math.150(3) (2016), 357–370] it was shown that the Schrödinger operator$H=-\unicode[STIX]{x1D6E5}+V$, with domain of definition$D(H)=C_{0}^{\infty }(\unicode[STIX]{x1D6FA})$and$V\in L_{\infty }^{\text{loc}}(\unicode[STIX]{x1D6FA})$, is essentially self-adjoint provided that$V(x)\geq (1-\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA}))/d(x)^{2}$. Here$d(x)$is the Euclidean distance to the boundary and$\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$is the nonnegative constant associated to the$L_{2}$-Hardy inequality. The conditions required for a domain to admit an$L_{2}$-Hardy inequality are well known and depend intimately on the Hausdorff or Aikawa/Assouad dimension of the boundary. However, there are only a handful of domains where the value of$\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$is known explicitly. By obtaining upper and lower bounds on the number of cubes appearing in the$k\text{th}$generation of the Whitney decomposition of$\unicode[STIX]{x1D6FA}$, we derive an upper bound on$\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x1D6FA})$, for$p>1$, in terms of the inner Minkowski dimension of the boundary.