Boundary behavior of Dirichlet eigenfunction of second order elliptic operators
We investigate the rate of decay of solutions of a second-order elliptic equation \(\tau f=\lambda f\) in an arbitrary domain \(\Omega \subset {\mathbb{R}}^ n\) which satisfy zero boundary conditions, and then apply the results to a study of the behavior of the associated heat semigroup in various weighted \(L^ 2\) spaces. Our results concern the behavior at infinity and at points on the boundary \(\partial \Omega\) of \(\Omega\), especially irregular ones like corners and cusps. The decay rates we obtain are expressed in terms of weighted \(L^ 2\) estimates which are governed by the sharpness of the cusps and the properties of the coefficients of \(\tau\). At a cusp we are able to estimate correctly the rate of decay in problems for which the answer is already known: as far as we know, rather general results fail to do this.
rate of decay of solutions of a second-order elliptic equation
heat semigroup in various weighted \(L^ 2\) spaces
boundary behavior of Dirichlet eigenfunctions
scientific article; zbMATH DE number 4107513
scientific article