  We prove a Li-Yau gradient estimate for positive solutions to the heat
equation, with Neumann boundary conditions, on a compact Riemannian submanifold
with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci
curvature assumption: \begin{equation} D^2 \sup_{x\in {\bf N}} \left(
\oint_{B(x,D)} |Ric^-|^p dy \right)^{\frac{1}{p}} < K \end{equation} for
$K(n,p)$ small enough, $p>n/2$, where $diam({\bf M})\leq D$. The boundary of
${\bf M}$ is not necessarily convex, but it needs to satisfy the interior
rolling $R-$ball condition.
