02.16.02 · analysis / sobolev-weak-solutions

Trace and Extension Theorems for Sobolev Functions

shipped3 tiersLean: none

Anchor (Master): Evans §5.4-§5.5; Adams-Fournier, Sobolev Spaces, 2e (Academic Press 2003), Ch. 5 (extension) and Ch. 7 (trace and fractional spaces); Nečas, Direct Methods in the Theory of Elliptic Equations (Springer 2012), Ch. 2; Lions-Magenes, Non-Homogeneous Boundary Value Problems and Applications I (Springer 1972), Ch. 1; Stein, Singular Integrals and Differentiability Properties of Functions (Princeton 1970), Ch. VI (the Stein extension operator)

Intuition Beginner

A function inside a region has values everywhere in the interior, but what is its value on the edge? If the function is continuous this is no puzzle: walk up to the boundary and read off the limit. The functions of partial differential equations are rougher. A typical Sobolev function is only defined up to changing it on a negligible set, and the boundary is itself a negligible set, so the raw values on the edge are pure ambiguity. The trace theorem rescues the situation: if you control the slope of the function inside, then the edge values are nonetheless well defined, in an averaged sense, as a genuine function living on the boundary.

Why should controlling the slope inside pin down the edge? Think of approaching the boundary along a short inward-pointing segment. The values along that segment are tied together by the slope: a function that does not change too fast cannot have a wild, undefined limit at the end of the segment. Averaging over all such approach segments turns the interior slope budget into honest control of the boundary reading. This averaged edge value is called the trace, and the trace theorem says it exists and is no larger, in total size, than the function and its slope inside.

There is a price. The boundary is one dimension thinner than the region, and squeezing a function onto a thinner set costs regularity: the trace is slightly rougher than the original. The exact accounting is that the trace loses a fraction of one derivative, a fraction set by the same exponent that governed the Sobolev inequalities. You hand in a function with one full derivative of control inside; you get back an edge function with a bit less than one derivative of control. This is the recurring theme of these theorems: restriction to a thinner set is allowed, but it is never free.

The companion result runs the other direction. The extension theorem says any function with controlled slope on a region can be continued to the whole surrounding space without losing control: there is a single mechanical recipe, the same for every function, that fills in the outside so the result still has a bounded slope everywhere. The trick is a mirror. Near a piece of flattened edge, you reflect the function across the edge, with a small correction so the slopes match up as you cross, and the reflected copy supplies the missing outside values.

A picture for both: imagine a damp cloth draped inside a bowl. The trace is the wet line the cloth leaves along the rim, well defined even though the cloth is rough, because the cloth cannot crumple too sharply. The extension is draping a matching second cloth over the outside of the bowl, joined smoothly at the rim, so the two together cover the whole table. Edge readings and edge continuations are the two halves of how rough functions meet the boundary.

Visual Beginner

The single picture to hold is a flattened boundary with an inward column and a mirror.

Read the picture top to bottom. Inside the region the function has a controlled slope. To find its edge value at a boundary point, approach along the short inward segment; the slope control forbids a wild limit, and averaging over nearby segments produces the trace, a bona fide function on the boundary that is slightly rougher than the original. That is the trace theorem.

Now read it as a mirror. To continue the function to the outside, reflect it across the flattened edge, adjusting the reflected copy so the slope does not jump as you cross. The two pieces glue into a single function on the whole space with a bounded slope everywhere. That is the extension theorem.

The side inset is the technical engine for both. Real boundaries are curved, but a smooth change of coordinates straightens any nice piece of boundary into a flat one, where the segment-averaging and the mirror are easy to carry out. Curved boundaries are handled one straightened patch at a time and then stitched back together.

Worked example Beginner

We watch the trace and the reflection extension work on the simplest region: the upper half-line setting, where the boundary is a single point and we track values approaching it. Take the region to be the numbers greater than zero, with boundary the point zero, and the function $u (x) = 2 - x$ for $x$ between zero and one (and tapering smoothly to zero beyond, which we ignore for the reading near the edge).

Step 1. Find the trace, the edge reading at zero. The function is continuous here, so we walk inward-to-edge: as $x$ shrinks toward zero, $u (x) = 2 - x$ approaches $2$ . The trace of $u$ at the boundary point zero is the number $2$ . The slope inside is constant and equal to minus one, a controlled slope, which is exactly what guarantees the edge reading is a clean number rather than garbage.

Step 2. Check the size accounting in spirit. The trace value $2$ is comparable to the size of the function near the edge, which sits between $1$ and $2$ on the unit segment. The edge reading did not blow up; it is bounded by the interior size and slope, the qualitative content of the trace bound.

Step 3. Now extend across the edge by mirror. We need to define $u$ for $x$ less than zero so the combined function has no slope jump at zero. The plain mirror would set $u (- x) = u (x)$ , giving the reflected value $2 - ∣ x ∣$ . Check the slopes: just right of zero the slope is minus one; just left of zero the plain mirror gives slope plus one. The slopes disagree, so the plain mirror has a kink at the edge.

Step 4. Correct the mirror to match slopes. Replace the plain mirror by a corrected one that flips the sign of the increment: define $u (x) = 2 + x$ for $x$ between minus one and zero. Now just left of zero the slope is plus one read as a function of the leftward direction, which lines up with the interior slope minus one as you pass through: the value is $2$ on both sides and the one-sided rates of change agree in magnitude, so the combined graph crosses the edge without a corner.

Step 5. Read the result. The extended function equals $2 - x$ to the right of zero and $2 + x$ to the left, a tent peaked at the value $2$ over the boundary point, with a bounded slope everywhere and no kink at the edge. The single mirror-with-correction recipe filled in the outside while keeping the slope under control.

What this tells us: the trace is just the edge limit made legitimate by slope control, and the extension is a mirror image glued on so the slope does not jump. Everything that follows at higher tiers is this same pair of moves, carried out for rough functions on curved boundaries by first straightening the boundary flat.

Check your understanding Beginner

Exercise (easy, multiple choice).

What does the trace of a Sobolev function record?

A. The largest value the function takes inside the region. B. The averaged boundary values of the function, a function living on the edge. C. The average of the function over the whole interior. D. The slope of the function at the centre of the region.

Hint

The trace is about the edge of the region, not the interior or the centre.

Answer

B. The averaged boundary values of the function, a function living on the edge. The trace assigns to an interior function a genuine function on the boundary, the edge reading made legitimate by slope control even when the raw boundary values are ambiguous. Feedback-correct: the trace lives on the boundary and is controlled by the interior size and slope. Feedback-wrong: A and C are interior quantities and D is a single number, none of which is a function on the edge.

Formal definition Intermediate+

Throughout, $n \geq 2$ , $1 \leq p < \infty$ , and $Ω \subseteq R^{n}$ is open and bounded. We assume $\partial Ω$ is $C^{1}$ (a Lipschitz boundary suffices for everything below, with surface measure interpreted via the coarea machinery 02.07.11); $ν$ denotes the outward unit normal and $d S$ the $(n - 1)$ -dimensional surface measure on $\partial Ω$ . The space $W^{1, p} (Ω)$ , its norm $∥ u ∥_{W^{1, p} (Ω)} = (∥ u ∥_{L^{p} (Ω)}^{p} + ∥ D u ∥_{L^{p} (Ω)}^{p})^{1/ p}$ , the weak gradient $D u = (D_{1} u, \dots, D_{n} u)$ , and the density of $C^{\infty} (\overset{ˉ}{Ω})$ in $W^{1, p} (Ω)$ for $C^{1}$ domains are taken as already established 24.01.01. We write $W_{0}^{1, p} (Ω)$ for the closure of $C_{c}^{\infty} (Ω)$ in $W^{1, p} (Ω)$ .

Definition (boundary chart / boundary flattening). A bounded domain $Ω$ has $C^{1}$ boundary if for each $x^{0} \in \partial Ω$ there are $r > 0$ and a $C^{1}$ function $γ : R^{n - 1} \to R$ with, after relabeling and reorienting axes, $Ω \cap B (x^{0}, r) = {x \in B (x^{0}, r) : x_{n} > γ (x_{1}, \dots, x_{n - 1})} .$ The map $Ψ (x) = (x_{1}, \dots, x_{n - 1}, x_{n} - γ (x_{1}, \dots, x_{n - 1}))$ is a $C^{1}$ diffeomorphism with $C^{1}$ inverse and Jacobian determinant identically $1$ ; it flattens the boundary, carrying $\partial Ω \cap B (x^{0}, r)$ into the hyperplane ${y_{n} = 0}$ and $Ω \cap B (x^{0}, r)$ into the half-ball ${y_{n} > 0}$ . Composition with $Ψ$ and $Ψ^{- 1}$ preserves $W^{1, p}$ with norms comparable up to a constant depending on $∥ D γ ∥_{L^{\infty}}$ (the chain rule, valid for weak derivatives because $γ \in C^{1}$ ).

Definition (trace operator). The trace operator is the bounded linear map $T : W^{1, p} (Ω) ⟶ L^{p} (\partial Ω)$ characterized by $T u = u ∣_{\partial Ω}$ whenever $u \in C^{\infty} (\overset{ˉ}{Ω}) \cap W^{1, p} (Ω)$ , extended to all of $W^{1, p} (Ω)$ by density. The defining bound is $∥ T u ∥_{L^{p} (\partial Ω)} \leq C ∥ u ∥_{W^{1, p} (Ω)}$ with $C = C (n, p, Ω)$ , proved in the Key Theorem; boundedness is exactly what makes the density extension well defined and independent of the approximating sequence ^{[Evans 2010 §5.5]}.

Definition (Sobolev-Slobodeckij / fractional trace space). For $0 < s < 1$ and $1 \leq p < \infty$ , the fractional Sobolev space $W^{s, p} (\partial Ω)$ consists of $g \in L^{p} (\partial Ω)$ with finite Gagliardo seminorm $[g]_{W^{s, p} (\partial Ω)}^{p} = \int_{\partial Ω} \int_{\partial Ω} \frac{∣ g ( x ) - g ( y ) ∣ ^{p}}{∣ x - y ∣ ^{(n - 1) + s p}} d S (x) d S (y),$ normed by $∥ g ∥_{W^{s, p} (\partial Ω)}^{p} = ∥ g ∥_{L^{p} (\partial Ω)}^{p} + [g]_{W^{s, p} (\partial Ω)}^{p}$ . The exponent $(n - 1) + s p$ in the denominator is the surface-dimension analogue of the $R^{n}$ Gagliardo seminorm from 02.16.01; it is the unique homogeneity making the seminorm scale like $s$ derivatives on the $(n - 1)$ -dimensional boundary.

Definition (extension operator). A (simple) bounded extension operator for $Ω$ is a bounded linear map $E : W^{1, p} (Ω) ⟶ W^{1, p} (R^{n}), E u = u a.e. on Ω, ∥ E u ∥_{W^{1, p} (R^{n})} \leq C ∥ u ∥_{W^{1, p} (Ω)},$ with $C = C (n, p, Ω)$ . The operator is simple when its norm bound is allowed to depend on $p$ ; the Calderón-Stein total extension operator is the stronger object that is bounded simultaneously on $W^{k, p} (Ω)$ for all $k$ and $1 \leq p \leq \infty$ with a single operator ^{[Stein 1970 Ch. VI]} ^{[Calderón 1961]}.

Counterexamples to common slips Intermediate+

The trace is not pointwise restriction. For $u \in W^{1, p} (Ω)$ the function is an equivalence class modulo null sets, and $\partial Ω$ is null in $R^{n}$ ; the symbol $u ∣_{\partial Ω}$ has no meaning before the trace theorem. $T$ is defined by continuity from smooth functions, and the bound is what gives the limit meaning. Two representatives of the same class have the same trace because they agree off a null set and $T$ is continuous.
Trace boundedness needs the full norm. The bound is $∥ T u ∥_{L^{p} (\partial Ω)} \leq C ∥ u ∥_{W^{1, p} (Ω)}$ with the full $W^{1, p}$ norm on the right, not the gradient seminorm alone: constants have zero gradient but nonzero trace, so a bare-gradient bound is impossible on a domain.
$L^{p} (\partial Ω)$ is not the sharp target. The trace lands in $L^{p} (\partial Ω)$ but does not fill it: the image is the strictly smaller fractional space $W^{1 - 1/ p, p} (\partial Ω)$ for $p > 1$ . The map onto $L^{p}$ is bounded but far from surjective; the map onto $W^{1 - 1/ p, p}$ is bounded and onto.
Extension requires boundary regularity. On a domain with an outward cusp the simple reflection extension fails, and for sufficiently sharp cusps no bounded extension operator exists at all: the inward geometry near the cusp lets a $W^{1, p} (Ω)$ function be too singular to continue. Lipschitz regularity of $\partial Ω$ is the standard sufficient condition (Calderón-Stein); $C^{1}$ is more than enough.
Order-one reflection is not enough for $W^{2, p}$ . The plain even reflection $u (x^{'}, - x_{n}) := u (x^{'}, x_{n})$ preserves $W^{1, p}$ but breaks the second derivative across the boundary (a normal-derivative jump). Extending $W^{k, p}$ needs a $k$ -th order reflection, a signed combination of dilated copies $\sum_{j} c_{j} u (x^{'}, - j x_{n})$ with the $c_{j}$ solving a Vandermonde system that kills derivative jumps up to order $k$ .

Key theorem with proof Intermediate+

Theorem (trace theorem). Let $Ω \subseteq R^{n}$ be bounded with $C^{1}$ boundary and $1 \leq p < \infty$ . There is a bounded linear operator $T : W^{1, p} (Ω) \to L^{p} (\partial Ω)$ such that $T u = u ∣_{\partial Ω}$ for $u \in C^{\infty} (\overset{ˉ}{Ω}) \cap W^{1, p} (Ω)$ and $∥ T u ∥_{L^{p} (\partial Ω)} \leq C ∥ u ∥_{W^{1, p} (Ω)}, C = C (n, p, Ω)$ ^{[Gagliardo 1957]} ^{[Evans 2010 §5.5, Theorem 1]}.

Proof. By density of $C^{\infty} (\overset{ˉ}{Ω})$ in $W^{1, p} (Ω)$ 24.01.01 it suffices to prove the bound for $u \in C^{1} (\overset{ˉ}{Ω})$ and then extend by continuity.

Step 1 (the half-space estimate). Consider first $u \in C^{1} (\overset{ˉ}{R_{+}^{n}})$ with compact support, where $R_{+}^{n} = {x_{n} > 0}$ and the boundary is ${x_{n} = 0}$ . Write $x = (x^{'}, x_{n})$ . Fix $x^{'} \in R^{n - 1}$ . By the fundamental theorem of calculus in the normal direction, for any $x_{n} > 0$ , $∣ u (x^{'}, 0) ∣^{p} = - \int_{0}^{x_{n}} \frac{\partial}{\partial t} ∣ u (x^{'}, t) ∣^{p} d t + ∣ u (x^{'}, x_{n}) ∣^{p} \leq ∣ u (x^{'}, x_{n}) ∣^{p} + p \int_{0}^{\infty} ∣ u (x^{'}, t) ∣^{p - 1} ∣ D_{n} u (x^{'}, t) ∣ d t .$ Average the inequality over $x_{n} \in (0, 1)$ (so the first term integrates against $u$ over a unit slab) and integrate over $x^{'} \in R^{n - 1}$ : $\int_{R^{n - 1}} ∣ u (x^{'}, 0) ∣^{p} d x^{'} \leq \int_{{0 < x_{n} < 1}} ∣ u ∣^{p} d x + p \int_{R_{+}^{n}} ∣ u ∣^{p - 1} ∣ D_{n} u ∣ d x .$ Apply Young's inequality $ab \leq \frac{1}{p ^{'}} a^{p^{'}} + \frac{1}{p} b^{p}$ with $a = ∣ u ∣^{p - 1}$ , $b = ∣ D_{n} u ∣$ , so $∣ u ∣^{p - 1} ∣ D_{n} u ∣ \leq \frac{p - 1}{p} ∣ u ∣^{p} + \frac{1}{p} ∣ D_{n} u ∣^{p}$ . This bounds the last integral by $∥ u ∥_{L^{p} (R_{+}^{n})}^{p} + ∥ D_{n} u ∥_{L^{p} (R_{+}^{n})}^{p}$ up to constants, giving $∥ u (\cdot, 0) ∥_{L^{p} (R^{n - 1})}^{p} \leq C (∥ u ∥_{L^{p} (R_{+}^{n})}^{p} + ∥ D u ∥_{L^{p} (R_{+}^{n})}^{p}) = C ∥ u ∥_{W^{1, p} (R_{+}^{n})}^{p} .$ This is the trace bound on the flat boundary.

Step 2 (localize and flatten). Cover $\partial Ω$ by finitely many boundary balls $B (x^{i}, r_{i})$ , $i = 1, \dots, N$ , on each of which the boundary is the graph of a $C^{1}$ function $γ_{i}$ flattened by the diffeomorphism $Ψ_{i}$ of the formal-definition section. Choose a partition of unity ${ζ_{i}}_{i = 0}^{N}$ subordinate to the cover, with $ζ_{0}$ supported in the interior and $\sum_{i} ζ_{i} \equiv 1$ on $\overset{ˉ}{Ω}$ . For each $i \geq 1$ , the function $v_{i} = (ζ_{i} u) \circ Ψ_{i}^{- 1}$ is a $W^{1, p}$ function on the half-space supported near a flat boundary patch, so Step 1 applies: $∥ v_{i} (\cdot, 0) ∥_{L^{p}}^{p} \leq C ∥ v_{i} ∥_{W^{1, p} (R_{+}^{n})}^{p} .$

Step 3 (transfer back and sum). The flattening $Ψ_{i}$ is a $C^{1}$ diffeomorphism with bounded Jacobian, so it carries surface measure on the flat patch to surface measure on $\partial Ω \cap B (x^{i}, r_{i})$ with comparable constants, and carries the $W^{1, p}$ norm of $v_{i}$ back to that of $ζ_{i} u$ with constants depending only on $∥ D γ_{i} ∥_{L^{\infty}}$ . Hence $∥ ζ_{i} u ∥_{L^{p} (\partial Ω)}^{p} \leq C_{i} ∥ ζ_{i} u ∥_{W^{1, p} (Ω)}^{p} \leq C_{i} ∥ u ∥_{W^{1, p} (Ω)}^{p},$ the last step using $∣ ζ_{i} ∣ \leq 1$ and $∣ D (ζ_{i} u) ∣ \leq ∣ D ζ_{i} ∣ ∣ u ∣ + ∣ D u ∣$ with $D ζ_{i}$ bounded. Summing over $i = 1, \dots, N$ (the interior piece $ζ_{0} u$ vanishes on $\partial Ω$ ) and using $u ∣_{\partial Ω} = \sum_{i} ζ_{i} u ∣_{\partial Ω}$ gives $∥ u ∥_{L^{p} (\partial Ω)} \leq C ∥ u ∥_{W^{1, p} (Ω)}$ .

Step 4 (density). The bound holds for all $u \in C^{\infty} (\overset{ˉ}{Ω})$ . Given $u \in W^{1, p} (Ω)$ , pick $u_{m} \in C^{\infty} (\overset{ˉ}{Ω})$ with $u_{m} \to u$ in $W^{1, p} (Ω)$ . Then $∥ u_{m} ∣_{\partial Ω} - u_{ℓ} ∣_{\partial Ω} ∥_{L^{p} (\partial Ω)} \leq C ∥ u_{m} - u_{ℓ} ∥_{W^{1, p} (Ω)} \to 0$ , so $(u_{m} ∣_{\partial Ω})$ is Cauchy in $L^{p} (\partial Ω)$ ; define $T u$ as its limit. The limit is independent of the sequence by the same bound, and $∥ T u ∥_{L^{p} (\partial Ω)} = lim ∥ u_{m} ∣_{\partial Ω} ∥_{L^{p}} \leq lim C ∥ u_{m} ∥_{W^{1, p}} = C ∥ u ∥_{W^{1, p}}$ . $□$

Bridge. The trace bound is exactly the boundary face of the same fundamental-theorem-of-calculus argument that drove the Sobolev inequalities of 02.16.01: there one integrated a derivative along axes to recover the function in the bulk; here one integrates the normal derivative along a single inward segment to recover the function on the edge, and the loss of a fraction of a derivative is the foundational reason the sharp target is the fractional space $W^{1 - 1/ p, p} (\partial Ω)$ rather than $L^{p} (\partial Ω)$ . This is dual to the extension theorem proved in the Full proof set below: trace restricts to the thinner boundary and loses regularity, while extension continues to the fatter space and preserves it, and putting these together gives the exact sequence in which $W_{0}^{1, p} (Ω) = ker T$ is the kernel and $W^{1 - 1/ p, p} (\partial Ω)$ is the image. The trace builds toward the weak formulation of boundary-value problems, where the boundary datum of an elliptic equation is by definition a trace, and the result appears again in 02.16.04 as the device that gives the Dirichlet condition meaning for non-smooth solutions and underlies the Gauss-Green integration-by-parts formula.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Carry out the one-dimensional core of the trace estimate: for $u \in C^{1} ([0, \infty))$ with $u (x) \to 0$ as $x \to \infty$ and $1 \leq p < \infty$ , show $∣ u (0) ∣^{p} \leq p \int_{0}^{\infty} ∣ u (t) ∣^{p - 1} ∣ u^{'} (t) ∣ d t$ , and deduce $∣ u (0) ∣^{p} \leq ∥ u ∥_{L^{p} (0, \infty)}^{p} + ∥ u^{'} ∥_{L^{p} (0, \infty)}^{p}$ up to a constant.

Hint

Write $∣ u (0) ∣^{p} = - \int_{0}^{\infty} \frac{d}{d t} ∣ u (t) ∣^{p} d t$ and bound $\frac{d}{d t} ∣ u ∣^{p} = p ∣ u ∣^{p - 1} sgn (u) u^{'}$ in absolute value. Then use Young's inequality $ab \leq \frac{p - 1}{p} a^{p / (p - 1)} + \frac{1}{p} b^{p}$ .

Answer

Since $u (t) \to 0$ at infinity, $∣ u (0) ∣^{p} = - \int_{0}^{\infty} \frac{d}{d t} ∣ u (t) ∣^{p} d t$ . The chain rule gives $\frac{d}{d t} ∣ u ∣^{p} = p ∣ u ∣^{p - 1} sgn (u) u^{'}$ , so $\frac{d}{d t} ∣ u ∣^{p} \leq p ∣ u ∣^{p - 1} ∣ u^{'} ∣$ and $∣ u (0) ∣^{p} \leq p \int_{0}^{\infty} ∣ u (t) ∣^{p - 1} ∣ u^{'} (t) ∣ d t .$ Apply Young's inequality with conjugate exponents $p^{'} = p / (p - 1)$ and $p$ to $a = ∣ u ∣^{p - 1}$ , $b = ∣ u^{'} ∣$ : $∣ u ∣^{p - 1} ∣ u^{'} ∣ \leq \frac{p - 1}{p} (∣ u ∣^{p - 1})^{p / (p - 1)} + \frac{1}{p} ∣ u^{'} ∣^{p} = \frac{p - 1}{p} ∣ u ∣^{p} + \frac{1}{p} ∣ u^{'} ∣^{p}$ . Integrating, $∣ u (0) ∣^{p} \leq (p - 1) ∥ u ∥_{L^{p}}^{p} + ∥ u^{'} ∥_{L^{p}}^{p} \leq C (∥ u ∥_{L^{p}}^{p} + ∥ u^{'} ∥_{L^{p}}^{p})$ . This is the trace bound in one dimension; the half-space case is this estimate fibered over $x^{'} \in R^{n - 1}$ .

Exercise 4 (medium, symbolic).

Construct the order-one reflection extension on the half-space. For $u \in C^{1} (\overline{R_{+}^{n}})$ define $\overset{u}{ˉ} (x^{'}, x_{n}) = u (x^{'}, x_{n})$ for $x_{n} \geq 0$ and $\overset{u}{ˉ} (x^{'}, x_{n}) = u (x^{'}, - x_{n})$ for $x_{n} < 0$ . Show $\overset{u}{ˉ} \in W^{1, p} (R^{n})$ with $∥ \overset{u}{ˉ} ∥_{W^{1, p} (R^{n})} \leq 2^{1/ p} ∥ u ∥_{W^{1, p} (R_{+}^{n})}$ , and identify which derivative is continuous across ${x_{n} = 0}$ and which is not.

Hint

Even reflection makes $\overset{u}{ˉ}$ continuous across the boundary and makes the tangential derivatives $D_{i} \overset{u}{ˉ}$ ( $i < n$ ) even, hence continuous; the normal derivative $D_{n} \overset{u}{ˉ}$ is odd, so it changes sign across ${x_{n} = 0}$ . Check there is no singular (distributional) part along the boundary because $\overset{u}{ˉ}$ itself has no jump.

Answer

$\overset{u}{ˉ}$ is continuous across ${x_{n} = 0}$ since both pieces equal $u (x^{'}, 0)$ there, so no boundary singular term appears in its distributional gradient. For $i < n$ , $D_{i} \overset{u}{ˉ} (x^{'}, x_{n}) = (D_{i} u) (x^{'}, ∣ x_{n} ∣)$ is the even reflection of $D_{i} u$ , continuous across the boundary. For the normal direction, differentiating $u (x^{'}, - x_{n})$ gives $D_{n} \overset{u}{ˉ} (x^{'}, x_{n}) = - (D_{n} u) (x^{'}, - x_{n})$ for $x_{n} < 0$ : the odd reflection of $D_{n} u$ , which flips sign across ${x_{n} = 0}$ and is generally discontinuous there, but is still an honest $L^{p}$ function with no Dirac part (the jump is in the value of an $L^{p}$ function, not a measure). Each of $∣ \overset{u}{ˉ} ∣^{p}$ and $∣ D \overset{u}{ˉ} ∣^{p}$ integrates to twice its half-space integral, so $∥ \overset{u}{ˉ} ∥_{W^{1, p} (R^{n})}^{p} = 2∥ u ∥_{W^{1, p} (R_{+}^{n})}^{p}$ , giving the factor $2^{1/ p}$ . Thus even reflection is a bounded extension $W^{1, p} (R_{+}^{n}) \to W^{1, p} (R^{n})$ ; it fails for $W^{2, p}$ precisely because the odd normal derivative develops a jump, which a second differentiation turns into a boundary Dirac mass.

Exercise 5 (medium, numeric).

A higher-order reflection $E u (x^{'}, x_{n}) = \sum_{j = 1}^{k + 1} c_{j} u (x^{'}, - j x_{n})$ for $x_{n} < 0$ extends $W^{k, p} (R_{+}^{n})$ provided $\sum_{j = 1}^{k + 1} c_{j} (- j)^{m} = 1$ for $m = 0, 1, \dots, k$ . For $k = 1$ (extending $W^{2, p}$ , so two equations $m = 0, 1$ with $j = 1, 2$ ), solve for $c_{1}, c_{2}$ .

Hint

The equations are $c_{1} + c_{2} = 1$ (from $m = 0$ ) and $- c_{1} - 2 c_{2} = 1$ (from $m = 1$ , since $(- j)^{1} = - j$ ). Solve the linear system.

Answer

$c_{1} = 3, c_{2} = - 2$ . From $m = 0$ : $c_{1} + c_{2} = 1$ . From $m = 1$ : $- c_{1} - 2 c_{2} = 1$ . Add the first to the second: $- c_{2} = 2$ , so $c_{2} = - 2$ , and then $c_{1} = 1 - c_{2} = 3$ . The extension $E u (x^{'}, x_{n}) = 3 u (x^{'}, - x_{n}) - 2 u (x^{'}, - 2 x_{n})$ for $x_{n} < 0$ matches $u$ and its first normal derivative across ${x_{n} = 0}$ (so $E u \in C^{1}$ across the boundary in the relevant weak sense), making it a bounded extension on $W^{2, p}$ . This is the $k = 1$ case of the Vandermonde-style reflection underlying the general Calderón-Stein construction.

Exercise 6 (medium, symbolic).

Prove the Gauss-Green / integration-by-parts formula for Sobolev functions: if $Ω$ is bounded with $C^{1}$ boundary, $u \in W^{1, p} (Ω)$ , $v \in W^{1, p^{'}} (Ω)$ with $\frac{1}{p} + \frac{1}{p ^{'}} = 1$ , then $\int_{Ω} u D_{i} v d x = - \int_{Ω} (D_{i} u) v d x + \int_{\partial Ω} (T u) (T v) ν_{i} d S$ . Reduce to the smooth case by density.

Hint

For $u, v \in C^{\infty} (\overset{ˉ}{Ω})$ the classical divergence theorem applied to the vector field $uv e_{i}$ gives the formula with $u ∣_{\partial Ω}, v ∣_{\partial Ω}$ . Approximate $u, v$ in $W^{1, p}, W^{1, p^{'}}$ and pass to the limit, using continuity of the trace and Hölder's inequality.

Answer

For $u, v \in C^{\infty} (\overset{ˉ}{Ω})$ the product rule gives $D_{i} (uv) = u D_{i} v + (D_{i} u) v$ , and the classical divergence theorem applied to $uv e_{i}$ yields $\int_{Ω} D_{i} (uv) d x = \int_{\partial Ω} uv ν_{i} d S$ , hence $\int_{Ω} u D_{i} v d x = - \int_{Ω} (D_{i} u) v d x + \int_{\partial Ω} uv ν_{i} d S$ . Now take $u_{m} \to u$ in $W^{1, p}$ and $v_{m} \to v$ in $W^{1, p^{'}}$ with $u_{m}, v_{m} \in C^{\infty} (\overset{ˉ}{Ω})$ . The bulk integrals converge by Hölder: $∣ \int_{Ω} u_{m} D_{i} v_{m} - \int_{Ω} u D_{i} v ∣ \leq ∥ u_{m} - u ∥_{L^{p}} ∥ D_{i} v_{m} ∥_{L^{p^{'}}} + ∥ u ∥_{L^{p}} ∥ D_{i} v_{m} - D_{i} v ∥_{L^{p^{'}}} \to 0$ , and similarly for the other bulk term. The boundary integral converges because $T u_{m} \to T u$ in $L^{p} (\partial Ω)$ and $T v_{m} \to T v$ in $L^{p^{'}} (\partial Ω)$ by continuity of the trace, so $\int_{\partial Ω} (T u_{m}) (T v_{m}) ν_{i} d S \to \int_{\partial Ω} (T u) (T v) ν_{i} d S$ again by Hölder. The formula passes to the limit. The boundary term is built entirely from traces, which is why the trace theorem is the prerequisite for any weak integration-by-parts.

Exercise 7 (hard, symbolic).

Prove $W_{0}^{1, p} (Ω) \subseteq ker T$ : if $u \in W_{0}^{1, p} (Ω)$ then $T u = 0$ . (The reverse inclusion, completing $W_{0}^{1, p} (Ω) = ker T$ , is harder and may be cited.)

Hint

By definition $u$ is a $W^{1, p}$ -limit of functions $φ_{m} \in C_{c}^{\infty} (Ω)$ , each of which vanishes in a neighbourhood of $\partial Ω$ , hence has zero trace. Use continuity of $T$ .

Answer

Let $u \in W_{0}^{1, p} (Ω)$ . By definition there are $φ_{m} \in C_{c}^{\infty} (Ω)$ with $φ_{m} \to u$ in $W^{1, p} (Ω)$ . Each $φ_{m}$ has compact support inside the open set $Ω$ , so $φ_{m} \equiv 0$ on a neighbourhood of $\partial Ω$ ; being smooth up to the boundary with $φ_{m} ∣_{\partial Ω} = 0$ , its trace is $T φ_{m} = φ_{m} ∣_{\partial Ω} = 0$ . The trace operator is bounded, hence continuous: $∥ T u ∥_{L^{p} (\partial Ω)} = ∥ T u - T φ_{m} ∥_{L^{p} (\partial Ω)} \leq C ∥ u - φ_{m} ∥_{W^{1, p} (Ω)} \to 0$ . Therefore $T u = 0$ , i.e. $u \in ker T$ . The converse $ker T \subseteq W_{0}^{1, p} (Ω)$ requires showing a zero-trace function can be approximated by compactly supported smooth functions, via translating $u$ inward and mollifying near a flattened boundary; together the two inclusions give the characterization $W_{0}^{1, p} (Ω) = ker T$ ^{[Evans 2010 §5.5]}.

Exercise 8 (hard, symbolic).

Show the half-space trace lands in the fractional space: for $u \in C_{c}^{1} (\overline{R_{+}^{n}})$ and $p > 1$ , the boundary value $g = u (\cdot, 0)$ satisfies the Gagliardo bound $[g]_{W^{1 - 1/ p, p} (R^{n - 1})}^{p} \leq C ∥ D u ∥_{L^{p} (R_{+}^{n})}^{p}$ , identifying the sharp trace exponent $s = 1 - 1/ p$ .

Hint

For $x^{'}, y^{'} \in R^{n - 1}$ write $g (x^{'}) - g (y^{'})$ as an integral of $D u$ over a path that dips into the half-space to height $\sim ∣ x^{'} - y^{'} ∣$ , e.g. up from $(x^{'}, 0)$ to $(x^{'}, r)$ , across to $(y^{'}, r)$ , and back down, with $r = ∣ x^{'} - y^{'} ∣$ . Estimate each leg by Hölder and substitute into the Gagliardo double integral.

Answer

Fix $x^{'}, y^{'}$ and set $r = ∣ x^{'} - y^{'} ∣$ . Using the fundamental theorem of calculus along the three-leg path through height $r$ , $∣ g (x^{'}) - g (y^{'}) ∣ \leq \int_{0}^{r} ∣ D_{n} u (x^{'}, t) ∣ d t + \int_{0}^{1} ∣ D^{'} u (x^{'} + θ (y^{'} - x^{'}), r) ∣ r d θ + \int_{0}^{r} ∣ D_{n} u (y^{'}, t) ∣ d t$ , where $D^{'}$ is the tangential gradient. The middle leg over the slab at height $r$ contributes the dominant control. Insert into the Gagliardo seminorm $\iint \frac{∣ g ( x ^{'} ) - g ( y ^{'} ) ∣ ^{p}}{∣ x ^{'} - y ^{'} ∣ ^{(n - 1) + s p}} d x^{'} d y^{'}$ with $s = 1 - 1/ p$ , so the denominator exponent is $(n - 1) + (p - 1)$ . After applying Hölder in each leg (cost $r^{p / p^{'}} = r^{p - 1}$ from the path length raised to $p$ via Jensen) and integrating the height variable, the $r$ -powers cancel exactly: the numerator scales like $r^{p - 1}$ times an $L^{p}$ average of $∣ D u ∣^{p}$ over a slab of thickness $r$ , and the denominator's $r^{(n - 1) + (p - 1)}$ together with the $d x^{'} d y^{'}$ measure absorbs the surplus, leaving $\int_{R_{+}^{n}} ∣ D u ∣^{p}$ after Fubini. The exponent $s = 1 - 1/ p$ is forced as the unique value making the $r$ -homogeneity balance, exactly as the scaling argument forced $p^{*}$ in 02.16.01. Hence $[g]_{W^{1 - 1/ p, p}}^{p} \leq C ∥ D u ∥_{L^{p} (R_{+}^{n})}^{p}$ .

Advanced results Master

The trace and extension theorems organize a larger structure: the sharp identification of the trace image, the surjectivity of the trace via a right inverse, the Calderón-Stein total extension operator, the quantitative Gauss-Green calculus, and the exact-sequence picture that makes $W_{0}^{1, p}$ , $W^{1, p}$ , and the boundary space fit together. Each refines the two fundamental-theorem-of-calculus arguments of the Intermediate tier.

Theorem 1 (sharp trace image; Gagliardo 1957, Aronszajn 1955). Let $Ω$ be bounded with $C^{1}$ boundary and $1 < p < \infty$ . The trace operator maps $W^{1, p} (Ω)$ onto the fractional Sobolev space $W^{1 - 1/ p, p} (\partial Ω)$ , and there is a bounded linear right inverse (a lifting or extension of boundary data) $L : W^{1 - 1/ p, p} (\partial Ω) \to W^{1, p} (Ω)$ with $T L = id$ ^{[Gagliardo 1957]} ^{[Aronszajn 1955]} ^{[Lions-Magenes 1972]}. At $p = 1$ the trace maps $W^{1, 1} (Ω)$ onto $L^{1} (\partial Ω)$ (the fractional index $1 - 1/ p$ degenerates to $0$ ), and the lifting is the harmonic-extension-type Poisson construction. The boundary datum loses exactly $1/ p$ of a derivative going in and the lifting restores it going out; the two together give an exact short sequence $0 ⟶ W_{0}^{1, p} (Ω) ⟶ W^{1, p} (Ω) T W^{1 - 1/ p, p} (\partial Ω) ⟶ 0,$ split by $L$ , with kernel $W_{0}^{1, p} (Ω)$ (the characterization theorem of the Full proof set).

Theorem 2 (Calderón-Stein total extension; Calderón 1961, Stein 1970). For a bounded Lipschitz domain $Ω$ there is a single linear operator $E : L_{l oc}^{1} (Ω) \to L_{l oc}^{1} (R^{n})$ with $E u = u$ on $Ω$ that is bounded $W^{k, p} (Ω) \to W^{k, p} (R^{n})$ simultaneously for every $k \in N_{0}$ and every $1 \leq p \leq \infty$ ^{[Calderón 1961]} ^{[Stein 1970 Ch. VI]}. Calderón's construction uses the singular-integral/Sobolev-representation machinery for $1 < p < \infty$ ; Stein's reflection across the Lipschitz boundary by a kernel $ψ$ with $\int_{1}^{\infty} ψ (λ) d λ = 1$ and vanishing moments removes the $p$ -dependence and the order-dependence at once. The single operator handing all $(k, p)$ at once is what makes the extension method usable as a black box across the whole Sobolev scale; the finite-order reflections of Exercise 5 are its truncations.

Theorem 3 (Gauss-Green and the normal trace; Gauss, Green, Sobolev form). For $Ω$ bounded with $C^{1}$ boundary, $u \in W^{1, 1} (Ω)$ , the divergence theorem holds in the trace form $\int_{Ω} D_{i} u d x = \int_{\partial Ω} (T u) ν_{i} d S,$ and for a vector field $F \in W^{1, 1} (Ω; R^{n})$ , $\int_{Ω} div F d x = \int_{\partial Ω} (T F) \cdot ν d S$ . This is the integration-by-parts identity of Exercise 6 with $v \equiv 1$ . The refinement to vector fields with only $div F \in L^{p}$ (not the full gradient) requires the normal trace $F \cdot ν \in W^{- 1/ p, p} (\partial Ω)$ defined by duality through the formula itself; this weaker normal trace is the natural object for conservation laws and for the De Giorgi-Federer theory of sets of finite perimeter, where $\partial Ω$ is merely rectifiable and $ν$ is the measure-theoretic normal supplied by the coarea/area formulas 02.07.11.

Theorem 4 (compact trace and the Sobolev trace inequality). For $Ω$ bounded with $C^{1}$ boundary, $1 \leq p < n$ , the trace operator $T : W^{1, p} (Ω) \to L^{q} (\partial Ω)$ is bounded for $1 \leq q \leq p_{†} := \frac{( n - 1 ) p}{n - p}$ (the trace-critical exponent, the boundary analogue of $p^{*}$ from 02.16.01) and compact for $1 \leq q < p_{†}$ . The borderline $q = p_{†}$ is the trace Sobolev inequality $∥ T u ∥_{L^{p_{†}} (\partial Ω)} \leq C ∥ u ∥_{W^{1, p} (Ω)}$ , whose sharp constants and extremals (boundary bubbles) parallel the Talenti-Aubin theory and govern the Escobar-Yamabe problem of prescribing boundary mean curvature. Compactness below $p_{†}$ is the boundary Rellich-Kondrachov theorem and supplies the boundary-term control in the direct method for mixed and Neumann problems.

Theorem 5 (trace spaces interpolate; Lions-Magenes). The trace scale ${W^{1 - 1/ p, p} (\partial Ω)}$ sits inside the real-interpolation family between $L^{p} (\partial Ω) = W^{0, p}$ and $W^{1, p} (\partial Ω)$ : $W^{s, p} (\partial Ω) = (L^{p} (\partial Ω), W^{1, p} (\partial Ω))_{s, p}$ for $0 < s < 1$ ^{[Lions-Magenes 1972]}. Consequently the trace theorem is one instance of a general functorial statement: restriction to a codimension-one boundary is a bounded map between the corresponding interpolation spaces, losing exactly the codimension-times- $(1/ p)$ amount of smoothness, $1/ p$ here. The same accounting predicts the trace onto a codimension- $d$ submanifold loses $d / p$ derivatives, recovering $W^{1 - d / p, p}$ when $d / p < 1$ and failing (no trace) when $d / p \geq 1$ .

Synthesis. The trace and extension theorems are the foundational reason elliptic boundary-value problems have a weak theory at all, and the entire structure is generated by the same single principle as the Sobolev inequalities of 02.16.01: integrate a derivative to recover the function, then balance the homogeneity by scaling. The trace is exactly this with one inward-normal integration, losing $1/ p$ of a derivative; the extension is dual to it, a reflection that continues the function with no loss; and putting these together yields the split exact sequence of Theorem 1, whose kernel $W_{0}^{1, p} (Ω)$ and image $W^{1 - 1/ p, p} (\partial Ω)$ are the two halves of how a rough function meets its boundary. The central insight is that the loss of $1/ p$ of a derivative is not an artifact of the proof but a scaling invariant — this is exactly why the sharp target is fractional and why the codimension- $d$ generalization of Theorem 5 loses $d / p$ derivatives, and it is the foundational reason the boundary-critical exponent $p_{†} = (n - 1) p / (n - p)$ of Theorem 4 has the same scaling pedigree as the interior $p^{*}$ .

The Gauss-Green calculus of Theorem 3 is the bridge that turns these function-space facts into the integration-by-parts identity on which the weak formulation rests: the boundary integral is built from traces, the normal trace is built by duality from the same formula, and the whole apparatus appears again in 02.16.04 as the machinery that gives Dirichlet, Neumann, and Robin conditions meaning for solutions too rough to restrict pointwise. From Gagliardo's 1957 characterization to the Calderón-Stein total extension and the Lions-Magenes interpolation picture, the theory is one continuous refinement of a single fundamental-theorem-of-calculus estimate carried out across the boundary instead of through the bulk.

Full proof set Master

Proposition 1 (boundedness of the simple reflection extension). Let $Ω$ be bounded with $C^{1}$ boundary, $1 \leq p < \infty$ . There is a bounded linear operator $E : W^{1, p} (Ω) \to W^{1, p} (R^{n})$ with $E u = u$ a.e. on $Ω$ and $∥ E u ∥_{W^{1, p} (R^{n})} \leq C (n, p, Ω) ∥ u ∥_{W^{1, p} (Ω)}$ .

Proof. Take the boundary cover ${B (x^{i}, r_{i})}_{i = 1}^{N}$ , flattening maps $Ψ_{i}$ , and partition of unity ${ζ_{i}}_{i = 0}^{N}$ from the trace proof. For $u \in C^{\infty} (\overset{ˉ}{Ω})$ work patch by patch. On a flat patch, where $Ω$ is locally ${x_{n} > 0}$ , define the higher-order (here first-order suffices for $W^{1, p}$ ) reflection $\overset{v}{ˉ} (x^{'}, x_{n}) = v (x^{'}, x_{n})$ for $x_{n} \geq 0$ and $\overset{v}{ˉ} (x^{'}, x_{n}) = v (x^{'}, - x_{n})$ for $x_{n} < 0$ , where $v = (ζ_{i} u) \circ Ψ_{i}^{- 1}$ . As computed in Exercise 4, $\overset{v}{ˉ} \in W^{1, p} (R^{n})$ with $∥ \overset{v}{ˉ} ∥_{W^{1, p} (R^{n})} \leq 2^{1/ p} ∥ v ∥_{W^{1, p} (R_{+}^{n})}$ : the even reflection makes $\overset{v}{ˉ}$ continuous across ${x_{n} = 0}$ (no singular boundary term in $D \overset{v}{ˉ}$ ), the tangential derivatives reflect evenly and the normal derivative reflects oddly, each remaining $L^{p}$ . Transport back by $Ψ_{i}$ , a $C^{1}$ diffeomorphism with bounded Jacobian, preserving $W^{1, p}$ up to constants depending on $∥ D γ_{i} ∥_{L^{\infty}}$ . The interior piece $ζ_{0} u$ extends by zero. Summing, $E u := \sum_{i} \overline{(ζ_{i} u) \circ Ψ_{i}^{- 1}} \circ Ψ_{i}$ (with $ζ_{0} u$ extended by zero) satisfies $E u = u$ on $Ω$ and $∥ E u ∥_{W^{1, p} (R^{n})} \leq C \sum_{i} ∥ ζ_{i} u ∥_{W^{1, p} (Ω)} \leq C^{'} ∥ u ∥_{W^{1, p} (Ω)}$ , using $∣ D (ζ_{i} u) ∣ \leq ∣ D ζ_{i} ∣∣ u ∣ + ∣ D u ∣$ with bounded $D ζ_{i}$ . Linearity and the bound let $E$ extend from the dense class $C^{\infty} (\overset{ˉ}{Ω})$ to all of $W^{1, p} (Ω)$ by continuity. $□$

Proposition 2 (trace of an extension recovers the boundary value). With $E$ as in Proposition 1 and $T$ the trace operator, $T (E u) ∣_{\partial Ω}^{int} = T u$ , i.e. the interior trace of the extension agrees with the trace of $u$ .

Proof. For $u \in C^{\infty} (\overset{ˉ}{Ω})$ , $E u$ is continuous across $\partial Ω$ (the even reflection introduces no jump), so its restriction to $\partial Ω$ from inside equals $u ∣_{\partial Ω} = T u$ . Both $u \mapsto T (E u)$ and $u \mapsto T u$ are bounded linear maps $W^{1, p} (Ω) \to L^{p} (\partial Ω)$ that agree on the dense subspace $C^{\infty} (\overset{ˉ}{Ω})$ , hence agree on all of $W^{1, p} (Ω)$ by continuity. $□$

Proposition 3 (zero extension characterizes $W_{0}^{1, p}$ via trace). Let $u \in W^{1, p} (Ω)$ , $Ω$ bounded with $C^{1}$ boundary. Then $T u = 0$ if and only if the extension of $u$ by zero, $\tilde{u} := u 1_{Ω}$ , lies in $W^{1, p} (R^{n})$ with $D \tilde{u} = (D u) 1_{Ω}$ (no boundary singular term).

Proof. Suppose $T u = 0$ . For a test field $ϕ \in C_{c}^{\infty} (R^{n}; R^{n})$ , the Gauss-Green formula of Exercise 6 (with $v$ replaced by components of $ϕ$ , valid since $ϕ$ is smooth) gives $\int_{R^{n}} \tilde{u} div ϕ d x = \int_{Ω} u div ϕ d x = - \int_{Ω} D u \cdot ϕ d x + \int_{\partial Ω} (T u) (ϕ \cdot ν) d S = - \int_{R^{n}} (D u) 1_{Ω} \cdot ϕ d x,$ the boundary term vanishing because $T u = 0$ . By definition of the weak gradient this says $\tilde{u} \in W^{1, p} (R^{n})$ with $D \tilde{u} = (D u) 1_{Ω}$ . Conversely, if $\tilde{u} \in W^{1, p} (R^{n})$ with $D \tilde{u} = (D u) 1_{Ω}$ , the same Gauss-Green identity run backwards forces $\int_{\partial Ω} (T u) (ϕ \cdot ν) d S = 0$ for all $ϕ$ ; choosing $ϕ \cdot ν$ to run over a dense set of boundary functions gives $T u = 0$ . Combined with the standard fact that $\tilde{u} \in W^{1, p} (R^{n})$ supported in $\overset{ˉ}{Ω}$ can be approximated by $C_{c}^{\infty} (Ω)$ functions (translate inward and mollify), this yields ${T u = 0} = W_{0}^{1, p} (Ω)$ , the kernel characterization. $□$

Proposition 4 (failure of trace at codimension $\geq p$ ). For $1 \leq p < \infty$ there is $u \in W^{1, p} (B_{1})$ , $B_{1} \subset R^{n}$ , whose restriction to a point (codimension $n$ ) or to a low-dimensional subset of codimension $d \geq p$ has no well-defined trace.

Proof. Take $u (x) = ∣ x ∣^{- α}$ on $B_{1} \subset R^{n}$ with $0 < α < (n - p) / p$ , so $u \in W^{1, p} (B_{1})$ by the computation of 02.16.01 Exercise 6 (gradient $\sim ∣ x ∣^{- α - 1}$ , integrable to the $p$ -th power exactly under this constraint). As $x \to 0$ , $u (x) \to \infty$ , so no finite value can serve as the trace of $u$ at the origin: the point ${0}$ has codimension $n$ , and $n / p \geq 1$ precisely when $p \leq n$ , which is the regime where this singular family lives. More generally, restriction to a codimension- $d$ affine subspace $Σ$ fails to be bounded once $d / p \geq 1$ : the same power function singular along $Σ$ (replace $∣ x ∣$ by distance to $Σ$ ) lies in $W^{1, p}$ but is unbounded on every neighbourhood of $Σ$ within $Σ$ , so no $L^{p} (Σ)$ trace exists. This is the sharp threshold $d < p$ matching the surviving fractional index $1 - d / p > 0$ of Theorem 5. $□$

Connections Master

The deep Sobolev embedding theorems and the scaling principle that forces critical exponents are developed in 02.16.01; the trace theorem is the boundary face of the same fundamental-theorem-of-calculus argument, and its critical exponent $p_{†} = (n - 1) p / (n - p)$ (Theorem 4) is the boundary analogue of the interior $p^{*}$ , derived by the identical homogeneity balance. That unit owns the interior embeddings; this unit owns the boundary restriction and continuation theory.
The surface measure $d S$ on $\partial Ω$ , the coarea formula that integrates over level sets and boundary slices, and the measure-theoretic normal vector used for the normal trace of Theorem 3 on merely rectifiable boundaries are supplied by 02.07.11; the boundary-flattening change of variables transports surface measure between curved and flat patches with the Jacobian bookkeeping that area-formula provides.
The Sobolev space $W^{1, p} (Ω)$ itself, its norm, the weak gradient, the density of $C^{\infty} (\overset{ˉ}{Ω})$ , and the definition of $W_{0}^{1, p} (Ω)$ as the closure of $C_{c}^{\infty} (Ω)$ are established in 24.01.01; this unit proves the trace and extension theorems that 24.01.01 states in survey form, and the kernel characterization $W_{0}^{1, p} (Ω) = ker T$ sharpens the bare definition of the zero-boundary-value space given there.
The weak formulation of elliptic boundary-value problems in 02.16.04 takes the trace as the very definition of the Dirichlet boundary datum, uses the lifting $L$ of Theorem 1 to reduce inhomogeneous boundary data to the homogeneous case, and rests on the Gauss-Green integration-by-parts identity of Theorem 3; the existence theory there is unintelligible without the trace machinery proved here.
The extension operator is the standard device for transferring interior results to domains: the Rellich-Kondrachov compactness and the full embedding ladder of 02.16.01, proved on $R^{n}$ , are pushed to a bounded domain $Ω$ by extending, applying the whole-space theorem, and restricting, so this unit's Proposition 1 is the bridge that makes the domain versions of the variational existence results in 02.16.05 available.

Historical & philosophical context Master

The notion that a function with finite Dirichlet integral has well-defined boundary values, despite being defined only up to null sets, was made precise by Nachman Aronszajn in his 1955 Kansas technical report on boundary values of functions with finite Dirichlet integral ^{[Aronszajn 1955]}, and independently and in sharp form by Emilio Gagliardo in his 1957 Rendiconti di Padova paper ^{[Gagliardo 1957]}, which identified the trace image of $W^{1, p} (Ω)$ as the fractional space now written $W^{1 - 1/ p, p} (\partial Ω)$ and constructed the bounded right inverse. The companion fractional spaces had been introduced by Lev Slobodeckij in 1958 ^{[Slobodeckij 1958]} through the Gagliardo seminorm, giving the scale its alternative name Sobolev-Slobodeckij.

The extension problem has an older and partly independent lineage. The reflection method for continuing a function across a flat boundary is classical, but the construction of a single operator bounded across the whole Sobolev scale was achieved by Alberto Calderón in his 1961 Berkeley symposium paper using the singular-integral representation of Sobolev functions ^{[Calderón 1961]}, and brought to its definitive Lipschitz-boundary form by Elias Stein in his 1970 monograph ^{[Stein 1970]}, whose total extension operator is bounded simultaneously on every $W^{k, p}$ for all orders and all exponents at once. Vasilii Babich had earlier given an extension construction for smooth domains in 1953 ^{[Babich 1953]}. The systematic functional-analytic organization of trace and lifting maps into the interpolation-space framework, with the boundary-value theory of elliptic and parabolic equations built on top, is due to Jacques-Louis Lions and Enrico Magenes in their 1968-1972 treatise ^{[Lions-Magenes 1972]}.

Bibliography Master

@article{Gagliardo1957,
  author  = {Gagliardo, Emilio},
  title   = {Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili},
  journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
  volume  = {27},
  year    = {1957},
  pages   = {284--305}
}

@article{Aronszajn1955,
  author  = {Aronszajn, Nachman},
  title   = {Boundary values of functions with finite Dirichlet integral},
  journal = {Conference on Partial Differential Equations, University of Kansas, Technical Report},
  volume  = {14},
  year    = {1955},
  pages   = {77--94}
}

@article{Slobodeckij1958,
  author  = {Slobodecki\u{\i}, Lev N.},
  title   = {Generalized Sobolev spaces and their application to boundary value problems for partial differential equations},
  journal = {Leningradskii Gosudarstvennyi Pedagogicheskii Institut. Uchenye Zapiski},
  volume  = {197},
  year    = {1958},
  pages   = {54--112}
}

@incollection{Calderon1961,
  author    = {Calder\'on, Alberto P.},
  title     = {Lebesgue spaces of differentiable functions and distributions},
  booktitle = {Partial Differential Equations},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {4},
  publisher = {American Mathematical Society},
  year      = {1961},
  pages     = {33--49}
}

@book{Stein1970,
  author    = {Stein, Elias M.},
  title     = {Singular Integrals and Differentiability Properties of Functions},
  publisher = {Princeton University Press},
  series    = {Princeton Mathematical Series},
  volume    = {30},
  year      = {1970}
}

@book{LionsMagenes1972,
  author    = {Lions, Jacques-Louis and Magenes, Enrico},
  title     = {Non-Homogeneous Boundary Value Problems and Applications, Vol. I},
  publisher = {Springer},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {181},
  year      = {1972}
}

@article{Babich1953,
  author  = {Babich, Vasilii M.},
  title   = {On the extension of functions},
  journal = {Uspekhi Matematicheskikh Nauk},
  volume  = {8},
  number  = {2},
  year    = {1953},
  pages   = {111--113}
}

@book{Necas2012,
  author    = {Ne\v{c}as, Jind\v{r}ich},
  title     = {Direct Methods in the Theory of Elliptic Equations},
  publisher = {Springer},
  series    = {Springer Monographs in Mathematics},
  year      = {2012}
}

Prerequisites

02.16.01
02.07.11
24.01.01

Tier anchors

beginner: Strogatz-style intuition for boundary values: why a function with controlled slope has a well-defined edge reading even when its raw values on the edge are ambiguous; Tao's blog discussion of restriction to lower-dimensional sets as a controlled loss of regularity
intermediate: Evans, Partial Differential Equations, 2e (AMS GSM 19, 2010), §5.4 (extensions) and §5.5 (traces); Brezis, Functional Analysis, Sobolev Spaces and PDEs (Springer 2011), §9.2-§9.4
master: Evans §5.4-§5.5; Adams-Fournier, Sobolev Spaces, 2e (Academic Press 2003), Ch. 5 (extension) and Ch. 7 (trace and fractional spaces); Nečas, Direct Methods in the Theory of Elliptic Equations (Springer 2012), Ch. 2; Lions-Magenes, Non-Homogeneous Boundary Value Problems and Applications I (Springer 1972), Ch. 1; Stein, Singular Integrals and Differentiability Properties of Functions (Princeton 1970), Ch. VI (the Stein extension operator)

References

Gagliardo — Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili · Rendiconti del Seminario Matematico della Università di Padova 27 (1957), 284-305
Slobodeckij — Generalized Sobolev spaces and their application to boundary value problems for partial differential equations · Leningradskii Gosudarstvennyi Pedagogicheskii Institut. Uchenye Zapiski 197 (1958), 54-112
Aronszajn — Boundary values of functions with finite Dirichlet integral · Conference on Partial Differential Equations, University of Kansas Technical Report 14 (1955), 77-94
Calderón — Lebesgue spaces of differentiable functions and distributions · Proceedings of Symposia in Pure Mathematics 4 (1961), 33-49
Stein — Singular Integrals and Differentiability Properties of Functions · Princeton University Press (1970), Ch. VI (the Stein total extension operator)
Babich — On the extension of functions · Uspekhi Matematicheskikh Nauk 8 (1953), no. 2, 111-113
Lions-Magenes — Non-Homogeneous Boundary Value Problems and Applications, Vol. I · Springer Grundlehren 181 (1972), Ch. 1
Nečas — Direct Methods in the Theory of Elliptic Equations · Springer Monographs in Mathematics (2012), Ch. 2
Evans — Partial Differential Equations, 2e · AMS Graduate Studies in Mathematics 19 (2010), §5.4-§5.5
Adams-Fournier — Sobolev Spaces, 2e · Academic Press (2003), Ch. 5, Ch. 7
Brezis — Functional Analysis, Sobolev Spaces and Partial Differential Equations · Springer Universitext (2011), §9.2-§9.4

Estimated time

beginner: 25m
intermediate: 65m
master: 105m