Counting dimensions of L-harmonic functions with exponential growth

Abstract

Let $\Omega \subset {\mathbb {R}}^{n-1}$ be a bounded open set, $X=\Omega \times {\mathbb {R}}\subseteq {\mathbb {R}}^{n}$ be the infinite strip. Let L be a second order uniformly elliptic operator of divergence form acting on a function $f\in W_{\text {loc}}^{1,2}(X)$ given by $Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )$. It is natural to consider the solutions of $Lu=0$ with boundary value $u|_{\partial \Omega \times {\mathbb {R}}}=0$ and exponential growth at most d: $|u(x',x_{n})|\le {\tilde{C}}e^{d|x_{n}|}$ for some ${\tilde{C}}>0$. Denote by ${\mathcal {A}}_{d}$ the solution space. In (Acta Math Sin (Engl Ser)15:525–534, 1999), Hang and Lin proved that $\text {dim}{\mathcal {A}}_{d}\le Cd^{n-1}$. The power $n-1$ is sharp, but one may wonder whether there are more precise estimates for the constant C. In this note, we consider some natural subspaces of ${\mathcal {A}}_{d}$ and obtain some estimates of dimensions of these subspaces. Compared with the case $L=\Delta _{X}$, when d is sufficiently large, the estimates obtained in this note are sharp both on the power $n-1$ and the constant C.