W2,p Estimates for the Monge—Amperè Equation

Our purpose in this chapter is to prove Caffarelli’s interiorLPestimates for second derivatives of solutions to the Monge—Ampère equation. That is, solutionsutoMu = fwithfpositive and continuous have second derivatives inLP,for 0 <p <∞, Theorem 6.4.2. The origin of these estimates goes back to Pogorelov [Pog71] who proved that convex solutions to detD2u = 1on a bounded convex domain Q withu =0 onasatisfy theL∞estimate 6.0.1$$ {C_1}(\Omega ',\Omega )Id \leqslant {D^2}u(x) \leqslant {C_2}(\Omega ',\Omega )Id$$, forxE Q’, where Q’ is a convex domain with closure contained in S2,Id isthe identity matrix, andCiare positive constants depending only on the domains. The estimates (6.0.1) have been proved in Chapter 4, and they follow as a consequence of Lemma 4.1.1; see (4.2.6).