One-sided control of the flat W^{1,p} trace #
For 1 ≤ p < ∞, the trace of a whole-space W^{1,p} function u on {a} × E is controlled by
its Sobolev norm on the half-space H = (a, ∞) × E, not just by its whole-space norm:
‖Tr u‖ₚᵖ ≤ (p - 1) ‖u‖_{Lᵖ(H)}ᵖ + ‖∇u‖_{Lᵖ(H)}ᵖ
(TauCeti.W1p.norm_hyperplaneTrace_rpow_le_restrictL). For p = 2 the right side is the squared
H¹(H) norm (TauCeti.W1p.norm_hyperplaneTrace_le_norm_restrictL). Consequently two whole-space
extensions of the same Sobolev function on this half-space have the same boundary trace. This is
the extension-independence step in constructing traces on domains with flat boundary, including
local boundary charts of strips.
The Euclidean product norm is represented by WithLp 2. The estimate is first proved on
test functions using the one-sided trace inequality
TauCeti.integral_hyperplane_abs_rpow_le_integral_halfSpace, then extended by whole-space Sobolev
density. This one-sided estimate bounds the trace TauCeti.W1p.halfSpaceTrace of arbitrary
half-space Sobolev functions, which is in TauCeti.Analysis.Sobolev.Trace.Extension together with
the reflection extension operator TauCeti.W1p.extendByReflectionL.
The argument follows L. C. Evans, Partial Differential Equations, Chapter 5, §5.5.
One-sided control of the flat trace. For 1 ≤ p < ∞, the p-th power of the Lᵖ norm
of the flat trace of u ∈ W^{1,p}(ℝ × E) is at most (p - 1) ‖u‖ᵖ + ‖∇u‖ᵖ, with both norms
taken on the right-hand half-space alone.
The flat H¹ trace is controlled by the Sobolev norm on the right-hand half-space alone.
Whole-space extensions whose values agree almost everywhere on the right-hand half-space have the same flat trace. Equality of their gradients need not be assumed.