The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles
The author considers an elliptic second order differential operator \(L\) of the form \[ L = \tfrac{1}{2} \,\nabla \cdot a \nabla + b \nabla + V, \] where \(a = [a_{i j}]_{i,j= 1}^{d} \in C^{2,d} (\mathbb{R}^{d})\) is uniformly elliptic on \(\mathbb{R}^{d}, \;\;b = (b_{1},...,b_{\alpha}) \in C^{1,\alpha} (\mathbb{R}^{d})\) and \(V \in C^{\alpha} (\mathbb{R}^{d}) \;\;\;(\alpha \in (0,1)).\) For a smooth compact domain \(D\) in \(\mathbb{R}^{d}\) it is imposed on \(\partial D\) either Dirichlet or an oblique-derivative boundary condition. The main purpose of the paper is to give asymptotic formulas for the principal eigenvalue of \(L\) under hard and soft obstacle type perturbations.
scientific article; zbMATH DE number 5508739