Estimates of \(C^m\)-capacity of compact sets in \(\mathbb R^N\)
Let \(L\) be a homogeneous elliptic partial differential operator with constant complex coefficients, and let \(V\) be a Banach space of distributions on \(\mathbb{R}^n\). This paper studies, for compacts \(X\) in \(\mathbb{R}^N\), the quantity \(\lambda_{V,L}(X)\), defined as the distance in \(V\) between the class of functions \(f_0\) satisfying \(Lf_0= 1\) near \(X\) and the space of functions \(f\) satisfying \(Lf= 0\) near \(X\). In particular, the paper considers the case where \(V= BC^m\) (when \(m\in \mathbb{N}\) this is the space of all \(m\) times continuously differentiable functions \(f\) such that \(\max_{|\alpha|\leq m}\sup_{\mathbb{R}^N}|\partial^\alpha f|\) is finite, but non-integer \(m\) are also allowed). The author obtains upper and lower bounds for \(\lambda_{V,L}(X)\) in terms of metric properties of the compact set \(X\).
scientific article; zbMATH DE number 2121422