  Fix $R>0$ and let $B_R$ denote the ball of radius $R$ centered at the origin
in $R^d$, $d\ge2$. Let $D\subset B_R$ be an open set with smooth boundary and
such that $R^d-\bar D$ is connected, and let $$
L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial
x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} $$ be a second order elliptic
operator. Consider the following linear heat equation in the exterior domain
$R^d-\bar D$ with boundary flux: \begin{equation*} \begin{aligned} &L u=0 \
\text{in}\ R^d-\bar D;\\ &a\nabla u\cdot \bar n=-h\ \text{on}\ \partial D;\\
&u>0 \ \text{is minimal}, \end{aligned} \end{equation*} where $h\gneq0$ is
continuous, and where $\bar n$ is the unit inward normal to the domain
$R^d-\bar D$. The operator $L$ must possess a Green's function in order that a
solution $u$ exist. An important feature of the equation is that there is no a
priori bound on the supremum $\sup_{x\in R^d-\bar D}u(x)$ of the solution
exclusively in terms of the boundary flux $h$, the hyper-surface measure of
$\partial D$ and the coefficients of $L$; rather the geometry of $D\subset B_R$
plays an essential role. However, we prove that in the case that $L$ is a \it
symmetric operator\rm\ with respect to some reference measure, then \it outside
of\rm\ $\bar B_R$, the solution to \eqref{LHE} is uniformly bounded,
independent of the particular choice of $D\subset B_R$. The proof uses a
combination of analytic and probabilistic techniques.
