Sobolev functions above the graph of a Lipschitz function, and their trace #
Let φ : E → ℝ be continuous. In the Euclidean product ℝ × E, the region above its graph is
the open set
TauCeti.normalEpigraph φ = {x | φ x.snd < x.fst}, the strict epigraph of φ,
whose boundary is the graph {(φ y, y) | y ∈ E}. These regions are the local models of domains
with Lipschitz boundary: near each boundary point such a domain is, after a rotation, the region
above the graph of a Lipschitz function. The half-space TauCeti.normalHalfSpace 0 is the case
φ = 0.
The graph-flattening homeomorphism TauCeti.graphFlattening φ : (t, y) ↦ (t - φ y, y) is
measure-preserving and carries the region above the graph onto the half-space {0 < t}; its
inverse adds φ y to the normal coordinate. If φ is K-Lipschitz, both maps are
(1 + K)-Lipschitz (TauCeti.lipschitzWith_graphFlattening), so their derivatives, which exist
almost everywhere, have norm at most 1 + K. By TauCeti.W1p.compMeasurePreservingL,
precomposition with them is then a pair of mutually inverse
bounded operators between W^{1,p} of the region and W^{1,p} of the half-space
(TauCeti.W1p.normalEpigraphEquivL), and for p < ∞ both preserve zero boundary values.
For 1 ≤ p < ∞, composing with the half-space trace TauCeti.W1p.halfSpaceTrace gives the
trace on the graph, TauCeti.W1p.normalEpigraphTrace : W^{1,p}(normalEpigraph φ) →L[ℝ] Lᵖ(E), a
bounded operator of norm at most p ^ (1 / p) (1 + K), and at most 1 + K for p = 2. The graph
is parametrized by y ↦ (φ y, y), the image under the inverse flattening of the boundary
point (0, y) of the half-space, and the trace is measured by Lebesgue measure on E rather
than by surface measure on the graph. Its kernel is exactly W^{1,p}_0 of the region
(TauCeti.W1p.ker_normalEpigraphTrace), the characterization of the homogeneous Dirichlet
condition by the trace.
Main declarations #
TauCeti.W1p.normalEpigraphEquivL:W^{1,p}of the region is isomorphic toW^{1,p}of the half-space, withTauCeti.W1p.normalEpigraphEquivL_mem_w1p0Submodule_iff.TauCeti.W1p.normalEpigraphTrace: the trace on the graph, withTauCeti.W1p.norm_normalEpigraphTrace_le,TauCeti.W1p.opNorm_normalEpigraphTrace_le, theirH¹versionsTauCeti.W1p.norm_normalEpigraphTrace_two_leandTauCeti.W1p.opNorm_normalEpigraphTrace_two_le, andTauCeti.W1p.ker_normalEpigraphTrace.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §5.5, Theorem 1 (the trace theorem is proved by flattening the boundary locally).
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, §9.3
(flattening
C¹boundaries) and Lemma 9.2. - J. Nečas, Direct Methods in the Theory of Elliptic Equations, Chapter 2 (Sobolev spaces and traces on domains with Lipschitz boundary, by local flattening).
Flattening W^{1,p} of the region above a Lipschitz graph #
Flattening the region above a Lipschitz graph. If φ is K-Lipschitz, precomposition
with inverse graph flattening is an isomorphism from
W^{1,p} of the region above the graph of φ onto W^{1,p} of the half-space {0 < t}, with
inverse the precomposition with TauCeti.graphFlattening φ. Both directions have norm at most
1 + K (TauCeti.W1p.norm_normalEpigraphEquivL_le).
Instances For
Flattening multiplies the W^{1,p} norm by at most 1 + K.
The inverse of flattening multiplies the W^{1,p} norm by at most 1 + K.
Flattening has operator norm at most 1 + K.
The inverse of flattening has operator norm at most 1 + K.
The flattened function is u ∘ (graphFlattening φ).symm.
The weak gradient of the flattened function is given by the chain rule.
The inverse of flattening is v ∘ graphFlattening φ.
The weak gradient of the inverse of flattening is given by the chain rule.
Flattening and its inverse preserve zero boundary values.
The trace on the graph #
The trace on the graph of a Lipschitz function. For 1 ≤ p < ∞ and a K-Lipschitz φ,
the trace of u ∈ W^{1,p} of the region {φ y < t} above the graph is the half-space trace of the
flattened function u ∘ (graphFlattening φ).symm. It is an element of Lᵖ(E): the graph is
parametrized by y ↦ (φ y, y), the image under inverse flattening of the boundary point (0, y)
of the half-space.
Instances For
The trace inequality on a graph domain: for 1 ≤ p < ∞,
‖Tu‖_{Lᵖ} ≤ p ^ (1 / p) (1 + K) ‖u‖_{W^{1,p}}.
For 1 ≤ p < ∞, the trace on a graph domain has operator norm at most
p ^ (1 / p) (1 + K).
The H¹ trace inequality on a graph domain: ‖Tu‖_{L²} ≤ (1 + K) ‖u‖_{H¹}.
The H¹ trace on a graph domain has operator norm at most 1 + K.
The functions of zero trace on a graph domain are those of W^{1,p}_0.
The kernel of the trace on a graph domain is W^{1,p}_0. For 1 ≤ p < ∞, on the region
above the graph of a Lipschitz function, the trace W^{1,p} → Lᵖ(E) vanishes exactly on the
closure of the test functions.