On the strong maximum principle for a fractional Laplacian
Let \(\Omega\) be a bounded open connected set in \(\mathbb{R}^n\) with Lipschitz boundary, \(s\in (\frac{1}{2},1)\), \(\{e_k\}\) be the sequence of eigenfunctions of the Laplace operator \(-\Delta: H_0^2(\Omega)\rightarrow L^2(\Omega)\), and \(\{\lambda_k\}\) be the sequence of corresponding eigenvalues. The authors establish a strong maximum principle for the spectral Dirichlet Laplacian \((-\Delta)^s\), defined by \[(-\Delta)^su=\sum_{k\in \mathbb{N}}\lambda_{k}^s\langle e_k,u\rangle_{L^2(\Omega)}e_k\] for each \(u=L^2(\Omega)\) such that \(\sum_{k\in \mathbb{N}}u_k^2\lambda_k^2<\infty\), where \(u_k\) is the component of \(u\) along \(e_k\). In particular, the authors prove that every nonnegative function \(u\in L^1(\Omega)\) such that \((-\Delta)^su\) is a Radon measure on \(\Omega\) is almost everywhere equal to a quasi continuous function \(\tilde{u}\), and if \(\tilde{u}=0\) on a subset of \(\Omega\) with positive \(H^s\)-capacity and \(u\) satisfies \[(-\Delta)^su+au\geq 0 \ \ \ \text{a.e. in} \ \ \Omega,\] for some nonnegative function \(a\in L^1(\Omega)\), then \(u=0\) in \(\Omega\). Ingredients of the proof are a fractional version of the Poincaré's inequality and truncated functions.
scientific article; zbMATH DE number 7374896
scientific article