The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash

Abstract

In this chapter, as in Chapter 9, we shall consider elliptic differential operators of divergence type. In order to concentrate on the essential aspects and not to burden the proofs with too many technical details, in this chapter we shall omit all lower-order terms and consider only solutions of the homogeneous equation. Thus, we shall investigate (weak) solutions of

$$ Lu = \sum\limits_{i,j = 1}^d {\frac{\partial } {{\partial x^j }}\left( {a^{ij} (x)\frac{\partial } {{\partial x^i }}u(x)} \right)} = 0, $$

where the coefficients a^ij are (measurable and) bounded and satisfy an ellipticity condition. We thus assume that there exist constants 0 < λ ≤ Λ < ∞ with

$$ \lambda \left| \xi \right|^2 \leqslant \sum\limits_{i,j = 1}^d {a^{ij} } (x)\xi _i \xi _j $$

((12.1.1))

for all x in the domain of definition Ω of u and all ξ ∈ ℝ^d, and

$$ \mathop {\sup }\limits_{i,j,x} \left| {a^{ij} (x)} \right| \leqslant \Lambda . $$

.