The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders

J. Korean Math. Soc. Published online September 22, 2020

Lidan Wang, Lihe Wang, and Chunqin Zhou
Shanghai Jiao Tong University, University of Iowa

Abstract : In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of $Lu(\mathbf{x})=a_{ij}(\mathbf{x})D_{ij}u(\mathbf{x})+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$
or $Lu(\mathbf{x})=D_{i}(a_{ij}(\mathbf{x})D_{j}u(\mathbf{x}))+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term $f$ of the equation.

Keywords : Unbounded Cylinder, Aleksandrov Maximum Principle, Boundary Harnack Inequality