Uniformly Elliptic Equations in Nondivergence Form

Abstract

In this chapter we consider linear operators of the form

$$\displaystyle{Lu =\sum _{ i,j=1}^{n}a_{ ij}(x)D_{ij}u(x)}$$

where the coefficient matrix A(x) = (a _ij (x)) is symmetric and uniformly elliptic, that is

$$\displaystyle{\lambda \vert \xi \vert ^{2} \leq \langle A(x)\xi,\xi \rangle \leq \Lambda \vert \xi \vert ^{2},}$$

for all $\xi \in \mathbb{R}^{n}$ and $x \in \Omega \subset \mathbb{R}^{n}.$