Sobolev localization across a chart boundary #
For open sets U ≤ Ω and a smooth bounded multiplier ψ with bounded gradient, assume
Ω ∩ tsupport ψ ⊆ U. Multiplication by ψ followed by zero extension defines the bounded
operator TauCeti.W1p.localizeL : W^{1,p}(U) →L[ℝ] W^{1,p}(Ω). Its norm is at most 2 M
when both multiplier bounds are M, for every 1 ≤ p ≤ ∞.
This localizes a function from a boundary-chart intersection to the original domain. The
support condition avoids the artificial boundary of the chart, while allowing nonzero
values at the physical boundary of Ω. Restricting the result to U recovers ψ u,
and localizing the restriction of a function on Ω equals multiplication by ψ there.
These identities allow local boundary constructions to be assembled with a partition of
unity. For every 1 ≤ p ≤ ∞, localizing the restriction of a W^{1,p}_0(Ω) function to an open set
V lands in W^{1,p}_0(V) when V ∩ tsupport ψ ⊆ Ω and Ω ∩ tsupport ψ ⊆ V
(TauCeti.W1p.localizeL_restrictL_mem_w1p0Submodule). In particular a W^{1,p}_0(Ω) function
vanishing off a compact subset of V restricts to W^{1,p}_0(Ω ∩ V)
(TauCeti.W1p.restrictL_mem_w1p0Submodule_of_isCompact). Consequently, zero boundary values
carry over to any open set agreeing with Ω near such a compact set
(TauCeti.W1p.mem_w1p0Submodule_of_inter_eq_of_isCompact).
For any set with compact closure in Ω and finite p,
TauCeti.W1p.exists_zeroBoundaryLocalization supplies a bounded cutoff operator into a
zero-boundary Sobolev space on a finite-measure interior domain, preserving values on that set.
No Sobolev embedding inequality or comparison with the dimension is assumed.
The construction reuses TauCeti.W1p.contDiffSMulL and zero extension of the ambient
value-gradient jet; admissibility follows from
TauCeti.HasWeakFDerivOn.indicator_of_isClosed. The localization argument is the one in
L. C. Evans, Partial Differential Equations, 2nd ed., §5.4 (extension by boundary charts).
Multiply a Sobolev function on U by a smooth cutoff, then extend it by zero to Ω.
The relative support condition permits nonzero values on the physical boundary of Ω.
The operator is bounded for every 1 ≤ p ≤ ∞, including both endpoints.
Instances For
The ambient jet of a localized function is the zero extension of its cutoff product.
The value of a localized function is ψ u on U and zero outside U.
The weak gradient of localization is the zero extension of ψ ∇u + u ∇ψ.
Localization has the same norm as the cutoff product on the source domain.
The explicit cutoff bound ‖localizeL u‖ ≤ 2 M ‖u‖.
The localization operator has norm at most 2 M.
Restricting localization back to the source domain recovers multiplication by the cutoff.
Localizing the restriction of a Sobolev function equals its cutoff product on Ω.
Localizing a zero-boundary Sobolev function from Ω to V preserves zero boundary
values when the two domains agree wherever the cutoff can be nonzero.
Zero boundary values localize. Let u ∈ W^{1,p}_0(Ω) vanish almost everywhere
off a compact set K contained in an open set V. Then the restriction of u to Ω ∩ V lies in
W^{1,p}_0(Ω ∩ V): the new part Ω ∩ ∂V of the boundary stays away from K.
Zero boundary values transfer between domains that agree near the support. Let Ω, V
and U be open with Ω ∩ U = V ∩ U, and let u ∈ W^{1,p}_0(Ω) vanish almost everywhere
off a compact set K ⊆ U. Then every w ∈ W^{1,p}(V) that agrees with u on Ω and vanishes
elsewhere lies in W^{1,p}_0(V): restricted to Ω ∩ U the function u keeps zero boundary
values, and w is its extension by zero to V ⊇ Ω ∩ U.
For finite p, one cutoff realizes restriction on V as a bounded linear map into a
zero-boundary Sobolev space on a finite-measure interior domain.