A Liouville theorem for supersolutions of linear elliptic second-order partial differential equations

Abstract:

We study supersolutions of linear elliptic second-order partial differential equations of the form \begin{equation*} Lu:=\sum \limits _{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j}=0,\tag *{($\ast $)} \end{equation*} which are defined and measurable in the whole space ${\mathbb R}^n$, and which belong locally to a Sobolev-type function space associated with the operator $L$ defined in ${\mathbb R}^n$, $n\geq 2$. We assume that the coefficients $a_{ij}(x)$ of the operator $L$ are measurable, locally bounded and such that $a_{ij}(x)=a_{ji}(x)$, and that the quadratic form associated with the operator $L$ is positive-definite. We prove a Liouville theorem for supersolutions of ($\ast$) defined in ${{\mathbb R}^n}$, in terms of a capacity associated with the operator $L$. As well, we establish a sharp distance at infinity between any non-constant supersolution of ($\ast$) in ${{\mathbb R}^n}$ bounded below by a constant and this constant itself, also in terms of the capacity associated with the operator $L$.