Approximation by harmonic functions in the \(C^ 1\)-norm and harmonic \(C^ 1\)-content of compact subsets in \(\mathbb{R}^ n\) (Q1326071)

Let \(X\) be a compact set in \(\mathbb{R}^ n\) \((n\geq 2)\), let \(C^ 1(\mathbb{R}^ n)\) be normed in the usual way, let \(J(X)= \{g\in C^ 1 (\mathbb{R}^ n)\): \(g|_ X=0\), \(\nabla g|_ X=0\}\) and let \(C^ 1(X)= C^ 1 (\mathbb{R}^ n)/ J(X)\). Further, let \(h^ 1(X)\) denote the closure in \(C(X)\) (suitably normed) of the subspace of equivalence classes which contain a function harmonic on some neighbourhood of \(X\). The harmonic \(C^ 1\)-content of \(X\) is defined by \(\Lambda^ 1(X)= \inf\{\sup \{| \nabla(f(x)- | x|^ 2)|\): \(x\in \mathbb{R}^ n\}\}\), where the infimum is taken over those continuously differentiable functions \(f\) on \(\mathbb{R}^ n\) which are harmonic on some neighbourhood of \(X\). It is shown that \(\Lambda^ 1(X)=0\) if and only if \(C^ 1(X)= h^ 1(X)\). Also, \(\Lambda^ 1(X)\leq c_ n V^{1/n}\), where \(V\) is the Lebesgue measure of \(X\) and \(c_ n\) is a constant which depends only on \(n\). The formula given for \(c_ n\) is sharp only when \(n=2\). Finally, if \(X\) has a piecewise smooth boundary of surface measure \(S\), then it is shown that \(2n V/S\leq \Lambda^ 1(X)\).