Hyperplane traces of zero-boundary Sobolev functions #
For 1 ≤ p < ∞, a function in W^{1,p}_0(Ω) extends by zero to a whole-space W^{1,p}
function. Composing that extension with the whole-space hyperplane trace gives a canonical trace of
zero-boundary functions on every hyperplane. If the hyperplane misses Ω, this trace vanishes. In
particular, W^{1,p}_0 functions on a half-space have zero trace on its boundary hyperplane.
This is the flat-model inclusion W^{1,p}_0(Ω) ⊆ ker(trace). The reverse inclusion requires a
trace defined on all of W^{1,p}(Ω) and an approximation argument near the boundary.
Main declarations #
TauCeti.W1p0.hyperplaneTrace: trace after whole-space extension by zero;TauCeti.W1p0.norm_hyperplaneTrace_le: its norm is at mostp ^ (1 / p)times theW^{1,p}_0norm, andTauCeti.W1p0.norm_hyperplaneTrace_le_norm: forp = 2, at most theH¹₀norm;TauCeti.W1p0.hyperplaneTrace_apply_eq_zero_of_forall_not_memandTauCeti.W1p0.hyperplaneTrace_eq_zero_of_forall_not_mem: the trace vanishes when the hyperplane misses the domain.
References #
L. C. Evans, Partial Differential Equations, Chapter 5, Section 5.5.
The Lᵖ trace on {a} × E of a zero-boundary Sobolev function, for 1 ≤ p < ∞, obtained by
extending the function by zero to the whole space before taking the whole-space hyperplane
trace.
Instances For
Taking the hyperplane trace of a zero-boundary function means taking the whole-space trace of its extension by zero.
The trace norm is at most p ^ (1 / p) times the zero-boundary W^{1,p} norm.
The operator norm of the zero-boundary hyperplane trace is at most p ^ (1 / p).
The zero-boundary H¹ trace norm is at most the H¹₀ norm.
The operator norm of the zero-boundary H¹ trace is at most one.
If the hyperplane {a} × E misses Ω, a zero-boundary Sobolev function on Ω has zero trace
there. This is the pointwise form of the flat-model inclusion W^{1,p}_0(Ω) ⊆ ker(trace).
If the hyperplane {a} × E misses Ω, the hyperplane trace operator on W^{1,p}_0(Ω) is
zero.