On fine limits and curving-shaped limits (Q5929906)

Let \(D\) be a domain in \(\mathbb{R}^N\) and let \(B(x,r)\) denote the open ball in \(\mathbb{R}^N\) with centre \(x\) and radius \(r\). A function \(f:D\to (0,+ \infty)\) is called an SIH-function (SIH = scale invariant Harnack) if there exists an increasing function \(g:(0,1) \to[1, +\infty)\) such that \(g(s)\to 1\) as \(s\to 0\) and \((g(t/r))^{-1} u(x)\leq u(y)\leq g(t/r)u(x)\) whenever \(B(x,2r) \subset D\) and \(t:=|x-y|\leq r\). A subdomain \(V\) of \(D\) is said to be a curving cone at \(\zeta\in \partial D\) if \(\zeta\in \partial V\) and for any sequence \((x_k)\) in \(V\) with limit \(\zeta\) there exists a constant \(\beta>0\) such that \(B(x_k, \beta|x_k-\zeta |)\subset D\) for all \(k\). A function \(f\) on \(D\) is said to have a curve-shaped limit \(L\) at \(\zeta\) if there exists a curving cone \(V\) at \(\zeta\) and \(f(x)\to L\) as \(x\to\zeta\) along \(V\) for each such \(V\).NEWLINENEWLINENEWLINEIt is shown that if a topology \(\Phi\) on \(\mathbb{R}^N\) is finer than the Euclidean topology and satisfies a certain condition, then an SIH-function \(i\) that has a \(\Phi\)-fine limit \(L\) at \(\zeta\) also has a curve-shaped limit \(L\) at \(\zeta\).