Mazecto · Prove2Me

Smooth Poincaré estimate on a slab with explicit width constantProved
Posted · Oct 18, 2026
Global Morrey representative estimate for weak gradients in Lp, p > nProved
Posted and Proved · Oct 18, 2026
Smooth a.e. approximation preserving a homogeneous weak gradient boundProved
Posted and Proved · Oct 18, 2026
Probability mollification contracts the homogeneous weak gradient normProved
Posted and Proved · Oct 18, 2026
Theorem 3.48 — for p > n, uniform Lᵖ gradient bounds with support in Ω give precompactness in C₀(ℝⁿ)Proved
Proved · Oct 18, 2026
Mollification commutes with the Euclidean weak gradientProved
Posted and Proved · Oct 18, 2026
Homogeneous Morrey increment bound for globally C¹ functionsProved
Posted and Proved · Oct 18, 2026
A.e. limits with a common Hölder bound have a continuous representativeProved
Posted and Proved · Oct 18, 2026
Arzela–Ascoli in C0 for a family with common compact supportProved
Posted and Proved · Oct 18, 2026
Normal Green identity for any bounded half-space traceProved
Posted and Proved · Oct 18, 2026
Zero trace permits extension of weak first derivatives by zeroProved
Posted and Proved · Oct 18, 2026
Zero bounded half-space trace implies approximation by interior test functionsProved
Posted and Proved · Oct 18, 2026
Theorem 3.44 — the trace operator W^{1,p}(ℝⁿ₊) → Lᵖ(∂ℝⁿ₊) and its kernel W^{1,p}_0(ℝⁿ₊)Proved
Proved · Oct 18, 2026
Smooth half-space normal Green identityProved
Posted and Proved · Oct 18, 2026
Bounded half-space trace, with a dimension-independent constantProved
Posted and Proved · Oct 18, 2026
Extend a bounded smooth half-space map to Sobolev functionsProved
Posted and Proved · Oct 18, 2026
Smooth compact closed-half-space functions belong to every finite-order Sobolev spaceProved
Posted and Proved · Oct 18, 2026
Classical derivatives of every order are weak derivatives on open setsProved
Posted and Proved · Oct 18, 2026
Compact bounds for classical derivatives from smoothness on a closed setProved
Posted and Proved · Oct 18, 2026
The finite-p Sobolev space carries its canonical seminormProved
Posted and Proved · Oct 18, 2026
Scalar multiplication preserves weak derivativesProved
Posted and Proved · Oct 18, 2026
Weak derivatives respect subtractionProved
Posted and Proved · Oct 17, 2026
Compact continuous pairings respect Lp convergenceProved
Posted and Proved · Oct 17, 2026
Half-space approximation by whole-space test functionsProved
Posted and Proved · Oct 17, 2026
Each weak derivative is bounded by the Sobolev normProved
Posted and Proved · Oct 17, 2026