Local Sobolev traces in Lipschitz boundary charts #
A smooth cutoff supported inside a Lipschitz boundary chart lets a Sobolev function be
transported to the whole epigraph of the chart's graph. Its epigraph trace is the localized
boundary trace, parametrized by y ↦ e.symm (φ y, y) and measured using Lebesgue measure on
the tangential space E, not surface measure on the frontier of the original domain;
TauCeti.W1p.frontierTrace glues these local traces into a trace for surface measure.
W1p.lipschitzChartLocalizeL performs the cutoff, zero extension past the artificial chart
boundary, and rigid coordinate change. It has norm at most 2 M. For 1 ≤ p < ∞, composing
with the graph trace gives W1p.lipschitzChartTrace : W^{1,p}(Ω) →L[ℝ] Lᵖ(E), of norm at most
2 M p ^ (1 / p) (1 + L), and at most 2 M (1 + L) for p = 2. Functions in W^{1,p}_0 have
zero local trace. Equality of the cutoff products suffices for equality of local traces. For a
function continuous up to the boundary, the local trace is the classical boundary value of the
cutoff product at the graph point (W1p.lipschitzChartTrace_apply_ae_of_continuousOn).
These are the local operators used when assembling a boundary trace by a partition of unity;
Set.IsLipschitzDomainWith.exists_lipschitzChart_cutoffs provides such a partition, with
cutoffs satisfying the hypotheses of W1p.lipschitzChartTrace, on a bounded Lipschitz domain.
Unlike reflection extension, taking a trace requires no shortening of the chart cylinder: the support need only lie inside the original cylinder.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §5.5, Theorem 1.
- J. Nečas, Direct Methods in the Theory of Elliptic Equations, Chapter 2 (traces on Lipschitz domains by local flattening).
The localized W^{1,p} trace through a Lipschitz boundary chart, for 1 ≤ p < ∞. The
result is in Lᵖ(E) with respect to tangential Lebesgue measure, not surface measure on
frontier Ω.
Instances For
The local chart trace is the epigraph trace of the localized function in chart coordinates.
The local trace estimate, with explicit cutoff and graph constants: for 1 ≤ p < ∞,
‖Tu‖_{Lᵖ} ≤ 2 M p ^ (1 / p) (1 + L) ‖u‖_{W^{1,p}}.
For 1 ≤ p < ∞, the local trace has operator norm at most 2 M p ^ (1 / p) (1 + L).
The local H¹ trace estimate, with explicit cutoff and graph constants:
‖Tu‖_{L²} ≤ 2 M (1 + L) ‖u‖_{H¹}.
The local H¹ trace has operator norm at most 2 M (1 + L).
A function with zero Sobolev boundary values has zero trace in every localized chart.
The local trace depends only on the cutoff product on the original domain. In particular, values outside the support of the cutoff have no effect on it.
The local trace of a function continuous up to the boundary. Let 1 ≤ p < ∞. If
u ∈ W^{1,p}(Ω) agrees almost everywhere with a function f continuous on closure Ω, then its
local chart trace is the classical boundary value of the cutoff product ψ f at the graph point
e.symm (φ y, y), for almost every y.
Chart cutoffs on a bounded Lipschitz domain. A bounded Lipschitz domain Omega has
finitely many Lipschitz boundary charts e i, φ i with radii a i, b i, smooth boundary
cutoffs ψ i bounded together with their gradients by M i and supported in the chart
cylinders, as TauCeti.W1p.lipschitzChartTrace requires, and a smooth interior cutoff ψ₀
compactly supported in Omega, such that ψ₀ + ∑ i, ψ i = 1 on the closure of Omega.