Zero extension of arbitrary-order Sobolev functions with zero boundary values #
Extension by zero from an open set Ω to a larger open set Ω' is a linear isometry
W^{k,p}_0(Ω) → W^{k,p}_0(Ω') for every natural order and 1 ≤ p ≤ ∞. Its value and
highest weak derivative are the zero extensions of the corresponding Lᵖ fields.
Restriction back to Ω recovers the original Sobolev function, and extensions compose.
No boundary regularity or boundedness is required.
This transport lets whole-space approximation and derivative estimates apply to zero-boundary functions on a domain. The zero-boundary condition is essential: a general domain Sobolev function can acquire singular distributional derivatives across the boundary.
The extension is characterized by continuity and its action on the dense family of test
functions: a test function on Ω becomes the same function on Ω'.
References #
L. C. Evans, Partial Differential Equations, §5.5; H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Lemma 9.5.
Extension by zero to a larger open set, as a linear isometry of zero-boundary
Sobolev spaces. This preserves the full norm at every order, including at p = ∞.
Equations
- TauCeti.Wkp0.extendByZeroₗᵢ hsub k = (TauCeti.Wkp0.ofTestFunctionₗ k ∘ₗ ↑(TestFunction.monoCLM ℝ)).extendOfIsometry ⋯ ⋯
Instances For
A test function extends to the same test function on the larger domain.
The value of the extension is the zero extension of the original value.
The highest recorded weak derivative extends by zero along with the value.
Zero extension commutes with forgetting the highest weak derivative.
Extending along the identity inclusion does nothing.
Zero extensions compose along inclusions of open sets.
Restricting the zero extension to its original domain recovers the Sobolev function.