The De Giorgi-Nash-Moser Estimates

with a positive constant λ. The equation Lu = 0 is then a second-degree elliptic equation. We also require the aij to be bounded and measurable, satisfying ‖aij‖L∞(B4) ≤ Λ with another constant Λ > 0. (In case you wonder, the radius ’4’ of the ball is to avoid fractions. Most of our estimates will be of the kind “some expression on B1 ≤ another expression on B4 ” and it would be unconvenient to have things like 1/8 as a radius.) It is clear that under these assumptions the equation (1) does not make sense, for the aij need not be differentiable. In fact, we shall use it as an abbreviation and really talk about so called weak solutions. Let us introduce these terms. A function u ∈ H(B4) is a weak solution to Lu = 0 if for all φ ∈ H 0 (B4), ∫ aijDjuDiφ = 0. (3)