A note on convergence of level sets
Given a sequence of functions \((f_n,f)\) defined on \(\mathbb{R}^N\) such that \(f_n\to f\) in some topology. The author states some convergence results in which the \(\delta_n\)-level set \(\Gamma_n:= \{x\in \mathbb{R}^N:|f_n(x)|\leq \delta_n\}\) of \(|f_n|\) converges in set theoretical sense to the \(0\)-level set \(\Gamma:= \{x\in\mathbb{R}^N: f(x)= 0\}\) of \(f\). There are two main results. The first result concerns with the case when \(f_n\), \(f\in L^p(\mathbb{R}^N)\) \((1\leq p<\infty)\) and \({\mathcal L}^{\mathbb{N}}\) is the Lebesgue measure on \(\mathbb{R}^N\). Let \(\varepsilon_n=\|f- f_n\|\to 0\) and \(0<\delta_n\to 0\) when \(n\to\infty\). (i) If \(\lim_{n\to\infty} \varepsilon_n/\delta_n= 0\), then \(\lim_{n\to\infty}{\mathcal L}^{\mathbb{N}}(\Gamma\Delta \Gamma_n)= 0\) and \({\mathcal L}^{\mathbb{N}}(\Gamma\Delta\limsup_{n\to \infty}\Gamma_n)= 0\); (ii) If \(\Sigma_n(\varepsilon_n/\delta_n)^p< \infty\), then \({\mathcal L}^{\mathbb{N}}(\Gamma\Delta\liminf_{n\to\infty} \Gamma_n)= 0\). Therefore \(\Gamma= \lim_{n\to\infty} \Gamma_n\) up to a set of \(0\)-Lebesgue measure. The second result concerns with the case when \(f_n\), \(f\in W^{1,p}(\mathbb{R}^N)\) \((1\leq p<\infty)\). Here \(\Gamma_n\) converges to \(\Gamma\) up to sets of \(0\)-capacity [see, e.g., \textit{L. C. Evans} and \textit{R. F. Gariepy}: ``Measure theory and fine properties of functions'' (1992; Zbl 0804.28001)]. Let \(f_n\), \(f\in W^{1,p}\), \(\varepsilon_n=\|f- f_n\|\to 0\) and \(0< \delta_n\to 0\) when \(n\to \infty\). (i) If \(\lim_{n\to\infty} \varepsilon_n/\delta_n= 0\), then \(\text{Cap}_p(\limsup_{n\to\infty} \Gamma_n\Delta\Gamma)= 0\); (ii) If \(\Sigma_n(\varepsilon_n/\delta_n)^p< \infty\), then \(\text{Cap}_p(\Gamma\Delta \liminf_{n\to\infty} \Gamma_n)= 0\). Therefore \(\Gamma= \lim_{n\to\infty} \Gamma_n\) up to a set of \(0\)-capacity.
scientific article; zbMATH DE number 1278882