On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

We investigate the vanishing of $H^1(X,{\mathcal{O}}_X(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,\mathrm{\Delta })$ over an algebraically closed field of positive characteristic $p$.