Uniformly Elliptic Equations in Nondivergence Form

In this chapter we consider linear operators of the form $$ Lu = \sum\limits_{i,j = 1}^n {{a_{ij}}(x){D_{ij}}u(x)} $$ where the coefficient matrix A(x) = (aij (x)) is symmetric and uniformly elliptic, that is $$\lambda {\left| \xi \right|^2} \leqslant \left\langle {A(x)\xi ,\xi } \right\rangle \leqslant \Lambda {\left| \xi \right|^2}$$, for all ξ∈ℝn and x∈Ω⊂ ℝn. We assume that the coefficients ai j are smooth functions, but the estimates we shall establish are independent of the regularity of the coefficients and depend only on the ellipticity constants λ, Λ and the dimension n.