Extension from bounded Lipschitz domains #
Every bounded Lipschitz domain in WithLp 2 (ℝ × E) admits a bounded linear extension
W^{1,p}(Ω) → W^{1,p}(ℝ × E) for every 1 ≤ p ≤ ∞. Restriction is a left inverse of this
operator. The construction uses a finite smooth partition of unity on the closure of the
domain, the local graph-reflection extension at boundary points, and zero extension of the
interior pieces. No differentiability of the boundary graphs is required.
As a consequence, restrictions to Ω of test functions on the whole space are dense in
W^{1,p}(Ω) for p < ∞ (TauCeti.W1p.denseRange_restrictL_ofTestFunctionₗ_top). These
restrictions are smooth up to the boundary, so a bounded operator on W^{1,p}(Ω) is determined
by its values on functions continuous on closure Ω.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §5.4, Theorem 1, for the partition-of-unity construction for smooth boundaries.
- L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised ed., §4.4, for extension across Lipschitz graphs.
Extension theorem for bounded Lipschitz domains. There is a bounded linear extension
from W^{1,p}(Ω) to the whole space, and restriction to Ω recovers the original function.
Both endpoint exponents p = 1 and p = ∞ are included.
Density of functions smooth up to the boundary. On a bounded Lipschitz domain and for
p < ∞, the restrictions to Ω of test functions on the whole space are dense in
W^{1,p}(Ω).