A vanishing result for strictly p-convex domains

In view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for (q) -complete domains in (mathbb C ^{n}) , we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly (p) -convex domains in (mathbb R ^n) in the sense of Harvey and Lawson (The foundations of (p) -convexity and (p) -plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the ({L}^2) -techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965).