Research proof index

A Campanato bound determines a Hölder representative up to a.e. equality.

In ℝ, the set of isolated points of any subset S is countable. Proof: cover by ⋃_{p,q ∈ ℚ} {unique element of S in (p,q)}. Each fiber has at most one element (uniqueness), and ℚ × ℚ is countable.

Crossover estimate bridging positive and negative powers of a positive supersolution.

Un-averaged half-ball version of crossover_estimate. This is the form needed by downstream weak-Harnack bookkeeping: multiply the average-product estimate by |B_{1/2}|^2.

Sobolev/Hölder step for De Giorgi pre-iteration on concentric balls. The argument passes through a zero-trace cutoff witness for η² (u - θ)₊. The originally intended statement with only the bare θ-truncation gradient on the right-hand side is false, e.g. for constant superlevel functions. The proof here records the actual Chapter 05 mechanism: 1. buil...

Sobolev/Hölder step for De Giorgi pre-iteration on concentric balls, using the concrete positive-part approximation input.