Sobolev extension through a Lipschitz boundary chart #
Let Ω be an open subset of the Euclidean product ℝ × E with a Lipschitz boundary chart: rigid
coordinates e identify Ω inside the cylinder normalCylinder a b with the region above the
graph of an L-Lipschitz function φ with φ 0 = 0. Let ψ be a smooth cutoff whose support
lies, in chart coordinates, in the shorter cylinder normalCylinder (a - 2 * L * b) b. This file
constructs a bounded operator
TauCeti.W1p.lipschitzChartExtensionL : W^{1,p}(Ω) →L[ℝ] W^{1,p}(ℝ × E)
for every 1 ≤ p ≤ ∞, whose value on Ω is ψ u
(TauCeti.W1p.restrictL_lipschitzChartExtensionL) and which vanishes outside the chart cylinder
(TauCeti.W1p.value_lipschitzChartExtensionL_ae_eq_zero). Its norm is at most 4 M (1 + L)²
when ψ and its gradient are bounded by M.
The construction localizes ψ u to the whole epigraph in chart coordinates
(TauCeti.W1p.lipschitzChartLocalizeL), extends across the graph by flattening and even
reflection (TauCeti.W1p.normalEpigraphExtensionL), and returns to the original coordinates.
The shorter normal radius of the support is what keeps the reflected function inside
the chart cylinder: over a tangential point y with ‖y‖ < b, the graph height satisfies
|φ y| ≤ L b, so the reflection t ↦ 2 φ y - t maps (-(a - 2 L b), a - 2 L b) into (-a, a).
Without it the reflected values could reach points of Ω outside the chart, and the extension
would not restrict to ψ u.
These local extensions are the boundary pieces of the classical extension operator for a bounded
Lipschitz domain: one sums them over a partition of unity subordinate to a finite chart cover
(Set.IsLipschitzDomainWith.exists_finset_isLipschitzChartWith_cover), together with the
extension by zero of an interior piece.
Main declarations #
TauCeti.W1p.lipschitzChartExtensionL: the extension operator through one chart.TauCeti.W1p.value_lipschitzChartExtensionL_ae: its value onΩisψ u.TauCeti.W1p.restrictL_lipschitzChartExtensionL: restricted toΩ, it isTauCeti.W1p.contDiffSMul ψ u.TauCeti.W1p.value_lipschitzChartExtensionL_ae_eq_zero: it vanishes outside the chart cylinder.TauCeti.W1p.norm_lipschitzChartExtensionL_le,TauCeti.W1p.opNorm_lipschitzChartExtensionL_le: the bound4 M (1 + L)².
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §5.4, Theorem 1 (extension by local
flattening, reflection and a partition of unity, for
C¹boundaries). - L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised ed.,
§4.4 (the
W^{1,p}extension for Lipschitz boundaries, by reflection across the local Lipschitz graph).
Reflection across the graph #
The open sets of the construction #
The extension operator #
Sobolev extension through a Lipschitz boundary chart. For a Lipschitz boundary chart of
Ω with coordinates e, graph φ and radii a, b, and a smooth cutoff ψ whose support has
chart coordinates in the shortened cylinder normalCylinder (a - 2 * L * b) b, the operator
sends u ∈ W^{1,p}(Ω) to a whole-space Sobolev function equal to ψ u on Ω
(TauCeti.W1p.restrictL_lipschitzChartExtensionL): multiply by the cutoff, extend across the
graph by reflection in flattened chart coordinates, and return to the original coordinates.
Instances For
The extension is supported in the chart cylinder. Outside the points whose chart
coordinates lie in normalCylinder a b, the extension vanishes almost everywhere.
The extension restricts to the cutoff product. On Ω, the value of the extension of
u is ψ u almost everywhere.
The extension is a right inverse of restriction up to the cutoff. Restricting the
extension of u to Ω gives the cutoff product ψ u.
The extension through a chart multiplies the W^{1,p} norm by at most 4 M (1 + L)².
The extension through a chart has operator norm at most 4 M (1 + L)².