Research proof index

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.

PDE-facing De Giorgi pre-iteration wrapper on concentric balls. This theorem packages the bookkeeping step that combines: - a localized De Giorgi energy estimate at level θ, - a Sobolev/Hölder interpolation input for level λ, - and the Chebyshev bound already proved in this chapter. The local Sobolev/Hölder interpolation theorem is kept as an explicit...

De Giorgi pre-iteration on concentric balls using the concrete positive-part witness constructor on the outer ball.

PDE-facing De Giorgi pre-iteration theorem on concentric balls. This now factors through the explicit cutoff-Sobolev bridge deGiorgi_cutoffSobolev_on_concentricBalls_of_ballPosPart instead of keeping the local Sobolev argument bundled into one monolithic proof.

Approximation-based version of the PDE-facing De Giorgi pre-iteration theorem. This is the approximation-driven counterpart of deGiorgi_preiter_on_concentricBalls_of_ballPosPart: instead of taking a pre-built witness for (u - θ)_+ or a generic positive-part closure axiom, it uses the outer-ball smooth approximation package for u - θ. The analytic pr...