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 ( ∗ ) L u := ∑ i , j = 1 n ( a i j ( x ) u x i ) x j = 0 , \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 R n {\mathbb R}^n , and which belong locally to a Sobolev-type function space associated with the operator L L defined in R n {\mathbb R}^n , n ≥ 2 n\geq 2 . We assume that the coefficients a i j ( x ) a_{ij}(x) of the operator L L are measurable, locally bounded and such that a i j ( x ) = a j i ( x ) a_{ij}(x)=a_{ji}(x) , and that the quadratic form associated with the operator L L is positive-definite. We prove a Liouville theorem for supersolutions of ( ∗ \ast ) defined in R n {{\mathbb R}^n} , in terms of a capacity associated with the operator L L . As well, we establish a sharp distance at infinity between any non-constant supersolution of ( ∗ \ast ) in R n {{\mathbb R}^n} bounded below by a constant and this constant itself, also in terms of the capacity associated with the operator L L .