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A smooth bump between concentric Euclidean balls

Statement

Let $0 < r < R$ and $n \geq 1$ . Then there exists a smooth function $ρ : R^{n} \to [0, 1]$ such that $ρ = 1$ on ${\overline{B}}_{r} (0)$ and $supp (ρ) \subseteq B_{R} (0)$ .

Facts & Assumptions

Given: Real numbers $0 < r < R$ .

[F1]

The standard smooth step function $σ$ is smooth, takes values in $[0, 1]$ , vanishes on $(- \infty, 0]$ , and equals $1$ on $[1, \infty)$ (The standard smooth step function).

[L1]

Coordinate differentiation uses the one-variable chain, sum and product rules (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ). Smoothness means that every iterated coordinate derivative exists and is continuous ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[F2]

The support is the closure of the nonzero locus (The support of a function on $R^{n}$ and its compactly supported Riemann integral).

Proof

technique · direct

1.1L1givenconstruct

Put $s = (r + R) / 2$ and $d = s^{2} - r^{2} > 0$ , and define $u (x) = (s^{2} - \sum_{j = 1}^{n} x_{j}^{2}) / d$ and $ρ (x) = σ (u (x))$ . The coordinate derivatives of $u$ are $\partial_{i} u = - 2 x_{i} / d$ , $\partial_{j} \partial_{i} u = - 2 δ_{i j} / d$ , and zero at every higher order, so $u$ is smooth. Here $δ_{i j}$ is $1$ when $i = j$ and $0$ otherwise.

2.1F1L1step 1.1

Every iterated coordinate derivative of $ρ$ is a finite sum of products of derivatives of $u$ and factors $σ^{(k)} (u)$ : induction follows by applying the chain and product rules in [L1] to each factor. These factors are continuous by [F1] and step 1.1, so $ρ$ is smooth.

2.2F1step 1.1

If $∥ x ∥ \leq r$ , then $u (x) \geq 1$ , so $ρ (x) = 1$ ; if $∥ x ∥ \geq s$ , then $u (x) \leq 0$ , so $ρ (x) = 0$ . Also $0 \leq ρ \leq 1$ by [F1].

3.1F2step 1.1step 2.1step 2.2∎

The nonzero locus lies in $B_{s} (0)$ , whose closure is contained in the closed ball ${\overline{B}}_{s} (0)$ . Thus [F2] gives $supp ρ \subseteq {\overline{B}}_{s} (0) \subseteq B_{R} (0)$ , since $s < R$ . Together with steps 2.1 and 2.2 this proves the claim.

Depends on

Used by

Compactly supported smooth functions are dense in W^k,p(Rⁿ) Corollary
Geodesics are exactly critical points of energy with fixed endpoints Corollary
Interior gradient bound for Poisson solutions Corollary
Weak differentiation has a closed graph on its natural domains Corollary
A hypersurface jump is not W^1,p Counterexample
A non-Dini continuous Poisson source can destroy the continuity of the second derivatives Counterexample
A nonzero boundary value creates a zero-extension jump Counterexample
A step has no locally integrable weak derivative Counterexample
Boundary H² regularity needs domain regularity Counterexample
Boundary W^2,p regularity needs more than Lipschitz boundary Counterexample
Bounded discontinuous elliptic coefficients need not give H² solutions Counterexample
Finite speed of propagation does not imply strong Huygens Counterexample
Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
Subcritical W^1,p is not closed under multiplication Counterexample
The strong Huygens principle in the homogeneous Cauchy setting Definition
A normalized compactly supported top form on Euclidean space Example
A point source produces a uniform expanding sphere Example
A radial bump on Euclidean space Example
A two-dimensional interior tail Example
Broken trajectories in an index-two torus moduli space Example
Dilation determines the Riesz-potential target exponent Example
L² forcing defines an H⁻¹ functional Example
Sharp Sobolev threshold for a radial power Example
The Morse complex of the circle Example
The Morse complex of the two-sphere Example
Compactly supported cohomology is contravariant for every smooth map False statement
A Euclidean bump for a compact set inside an open set Lemma
Caccioppoli inequality for truncated subsolutions Lemma
Compactly supported top cohomology propagates across overlapping oriented coordinate balls Lemma
Every integer is realized by a map to the sphere Lemma
Heat-ball representation formula Lemma
Local W^2,p regularity of weak solutions of the Poisson equation Lemma
Locally finite smooth partitions of unity on domains Lemma
Logarithmic Caccioppoli estimate for positive supersolutions Lemma
Moser iteration for positive supersolutions: negative-power and logarithmic comparison Lemma
Nondegenerate local Hölder coordinates for a Hölder coefficient Lemma
Peak functions at strongly pseudoconvex boundary points, by a dbar correction Lemma
Smooth compactly supported functions of an open set are dense in L² Lemma
Smooth finite point motions extend to disk isotopies Lemma
Sobolev level-set step: energy decay with explicit level gap and radius loss Lemma

…and 17 more results.

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)