Neumann Li-Yau gradient estimate under integral Ricci curvature bounds

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:

$D^2 \sup _{x\in {\bf N}} \left ( \oint _{B(x,D)} \vert Ric^-\vert^p dy \right )^{\frac {1}{p}} < K$

(1)

for $K(n,p)$ small enough, $p>n/2$, and $diam({\bf M})\leq D$. The boundary of ${\bf M}$ is not necessarily convex, but it needs to satisfy the interior rolling $R-$ball condition.