Asymptotic behavior of the ground state of singularly perturbed elliptic equations
Let \((M,g)\) be a smooth compact \(n\)--dimensional Riemannian manifold. In a certain coordinate system the metric is denoted by \(a_{ij}(x), a^{ij}(x):= (a_{ij}(x))^{-1}\). The author considers a family, \(A^\mu\), of second order elliptic operators which in local coordinates have the form \({1\over 4} \mu^2 \Delta +\mu b^i(x) {\partial \over \partial x^i} + v(x),\) where \(b^i,v\) are continuously differentiable real functions. The author also requires that \(a^{ij}(x)\) is continuously differentiable and the matrix \(a_{ij}\) is uniformly positive definite. However, the reviewer has the feeling that on a smooth compact manifold the metric tensor is smooth and thanks to the compactness the metric tensor is uniformly positive definite. The operator \(\mu^2\Delta\) is nothing but the Laplacian for the scaled metric \(\mu^{-2}g\). The author investigates the limiting behavior of the first eigenvalue, \(\lambda_0(\mu)\), and the first eigenfunction, \(p_0(\mu)\), as \(\mu\) tends to \(0\). He shows that \(\lambda_0(\mu)\) approaches a limit as \(\mu\to 0\). Furthermore, under the additional assumption that the family \(A^\mu\) is recursive, it is shown that \(\mu \ln p_0(x)\) approaches a limit, uniformly in \(x\in M\). Of special interest is the case where the first order term has a potential. Then the limit of the first eigenvalue can be expressed as a minimum of a certain expression in the potential. Also the case of self--adjoint operators \((b(x)=0)\) is discussed separately.
scientific article; zbMATH DE number 1316930