$W^{s,\frac {n}{s}}$-maps with positive distributional Jacobians

Abstract

We extend the well-known result that any $f \in W^{1,n}({\Omega }, \mathbb {R}^{n})$, ${\Omega } \subset \mathbb {R}^{n}$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac {n}{s}}({\Omega })$ for any $s \geq \frac {n}{n+1}$, where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac {n}{s}}$ of the degree estimates known for W^1,n-maps with positive or non-negative Jacobian, such as the sense-preserving property.

Appendix A: Fractional Sobolev spaces

We recall results from fractional Sobolev spaces we employ here. Most of them are well-known to most experts.

We begin with proving Lemma 1.3. Lemma 1.3 was (essentially) proven in [28] as an extension of the ground-breaking paper [9], which showed that Jacobians of W^1,n-maps can be tested with BMO-maps. The proof in [28] uses Littlewood-Paley theory and paraproducts. In [5] Brezis and Nguyen gave a simpler and more elegant proof of this result for $s =\frac {n}{n+1}$. We present here the following slight adaptation of their argument from [21].

We restrict our attention to the a priori estimates, from which the claim follows easily due to multi-linearity.

Proof of 1.3 (a priori estimates)

Let $\varphi \in C_{c}^{\infty }({\Omega })$ and $f \in C^{\infty }(\overline {{\Omega }})$. Ω is an extension domain, [20, 33], so we may assume that $f \in W^{s,\frac {n}{s}}(\mathbb {R}^{n}) \cap C^{1}(\mathbb {R}^{n})$. Then

$$ {\int}_{{\Omega}} \det(Df) \varphi = {\int}_{\mathbb{R}^{n}} \det(Df) \varphi. $$

Extend f and φ harmonically to $\mathbb {R}^{n+1}_{+}$, say to F and Φ respectively. We write $(x,t) \in \mathbb {R}^{n} \times \mathbb {R}_{+} = \mathbb {R}^{n+1}_{+}$. By Stokes’ theorem and Hölder’s inequality,

$$ \begin{array}{@{}rcl@{}} \left |{\int}_{\mathbb{R}^{n}} \det(Df) \varphi \right |&=& \left |{\int}_{\mathbb{R}^{n+1}_{+}} \det(DF|D{\Phi}) \right |\\&\leq& \left ({{\int}_{\mathbb{R}^{n+1}_{+}} |t^{1-\frac{s}{n}-s} DF|^{\frac{n}{s}}} \right )^{s} \left ({{\int}_{\mathbb{R}^{n+1}_{+}} |t^{1-(1-s)-(1-s)n}D{\Phi}|^{\frac{1}{1-s}}} \right )^{1-s}. \end{array} $$

If $s \in (\frac {n-1}{n},1]$, then (1 − s)n ∈ (0, 1). Then, by trace estimates, see e.g. [21, Proposition 10.2], we have

$$ \left ({{\int}_{\mathbb{R}^{n+1}_{+}} |t^{1-\frac{s}{n}-s} DF|^{\frac{n}{s}}} \right )^{s} \approx [f]_{W^{s,\frac{n}{s}}({{{\Omega}}})}^{n} $$

and

$$ \left ({{\int}_{\mathbb{R}^{n+1}_{+}} |t^{1-(1-s)-(1-s)n}D{\Phi}|^{\frac{1}{1-s}}} \right )^{1-s} \approx [\varphi]_{W^{(1-s)n,1-s}({{\Omega}})}. $$

Here we also used the fact that $[f]_{W^{s,\frac {n}{s}}(\mathbb {R}^{n})} \precsim [f]_{W^{s,\frac {n}{s}}({{\Omega })}}$. This is because Ω is an extension domain; see [20, 33]. We conclude, because we have shown

$$ {\int}_{{\Omega}} \det(Df) \varphi \precsim [f]_{W^{s,\frac{n}{s}}({\Omega})}^{n} [\varphi]_{W^{(1-s)n,1-s}({\Omega})}. $$

□

The ensuing result on trace operators will be useful for the subsequent developments. For detailed treatments we refer to [26, §2.4.2, Theorem 1], [1, Theorem 7.43, Remark 7.45] and [30, Lemma 36.1].

Lemma A.1 (Trace Theorem)

Let ${\Omega } \subset \mathbb {R}^{n}$ be either bounded or the complement of a bounded set, with smooth boundary. If s ∈ (0, 1), $p \in (1,\infty )$ with $s-\frac {1}{p} > 0$, then the trace operator on T = |_∂Ω is a bounded, linear, surjective operator from W^s,p(Ω) to $W^{s-\frac {1}{p},p}(\partial {\Omega })$. The harmonic extension is a bounded linear right-inverse of T.

The following is well-known for Sobolev functions in W^1,p (it is essentially Fubini’s theorem):

Lemma A.2 (Restriction theorem)

For Ω a smooth, bounded domain let f ∈ W^s,p(Ω). Fix x₀ ∈Ω. There exists a representative of f such that for ${\mathscr{L}}^{1}$-almost every r ∈ (0,dist (x₀, ∂Ω)) we have f ∈ W^s,p(∂B(x₀, r)).

Moreover, for Ω = B(x₀, R) we have

$$ \left( {{{\int}_{0}^{R}} [f]_{W^{s,p}(\partial B(x_{0},r))}^{p} dr}\right)^{\frac{1}{p}} \precsim [f]_{W^{s,p}(B(x_{0},R))}. $$

Proof

As Ω is an extension domain, see [20, 33], we may assume that ${\Omega } = \mathbb {R}^{n}$ and $f \in W^{s,p}(\mathbb {R}^{n})$ with f ≡ 0 outside a compact set. Denote by $F: \mathbb {R}^{n+1}_{+} \to \mathbb {R}$ the harmonic extension of f, and w.l.o.g. set x₀ = 0. Then (see [21, Proposition 10.2])

$$ \left \|(x_{n+1})^{1-\frac{1}{p}-s} DF \right \|_{L^{p}(\mathbb{R}^{n+1}_{+})} \approx [f]_{W^{s,p}(\mathbb{R}^{n})} < \infty. $$

By Fubini’s theorem, for ${\mathscr{L}}^{1}$-almost every r > 0,

$$ \left \|(x_{n+1})^{1-\frac{1}{p}-s} DF \right \|_{L^{p}(\partial B(r) \times (0,\infty))} < \infty. $$

This implies that f ∈ W^s,p(∂B(r)) for almost every r > 0.

The last claim also follows from Fubini’s theorem in $\mathbb {R}^{n+1}_{+}$:

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{R}} [f]_{W^{s,p}(\partial B(x_{0},r))}^{p} dr &\precsim& {{\int}_{0}^{R}} {\int}_{\partial B(x_{0},r) \times (0,\infty)} |(x_{n+1})^{1-\frac{1}{p}-s} DF|^{p} \\&=& {\int}_{B(x_{0},R) \times (0,\infty)} |(x_{n+1})^{1-\frac{1}{p}-s} DF|^{p}. \end{array} $$

□

Lemma A.3

Let ${\Omega } \subset \mathbb {R}^{n}$ be a bounded domain with smooth boundary. For s ∈ (0, 1), $p \in (1,\infty )$ such that $s-\frac {1}{p} > 0$:

• (1) If f ∈ W^s,p(Ω) and $g \in W^{s,p}(\mathbb {R}^{n} \backslash {\Omega })$ with f = g on ∂Ω in the trace sense. Then

  $$ h := \begin{cases} f \quad & \text{in} {\Omega}\\ g \quad &\text{in} \mathbb{R}^{n} \backslash {\Omega} \end{cases} $$

  belongs to the Sobolev space and

  $$ [h]_{W^{s,p}(\mathbb{R}^{n})} \leq C({\Omega}) \left( [f]_{W^{s,p}({\Omega})} + [g]_{W^{s,p}(\mathbb{R}^{n}\setminus{\Omega})}\right). $$

• (2) In particular, if f ∈ W^s,p(Ω) satisfies f = 0 on ∂Ω in the trace sense, then $f\in W^{s,p}_{0}({\Omega })$ and that

  $$ h := \begin{cases} f \quad &\text{in} {\Omega}\\ 0 \quad &\text{in} \mathbb{R}^{n} \backslash {\Omega} \end{cases} $$

  belongs to $W^{s,p}(\mathbb {R}^{n})$.

Lemma A.4

Let B(R) be a ball in $\mathbb {R}^{n}$. Let $f \in W^{s,\frac {n}{s}}(B(R))$ for some s ∈ (0, 1) and f|_∂B(R) ∈ C⁰(∂B(R)). Then there exists an approximation $f_{k} \in C^{\infty }_{c} (\mathbb {R}^{n})$ converging to f in $W^{s,\frac {n}{s}}(B(R))$ and $f_{k} \rightrightarrows f$ uniformly on ∂B(R).

Proof

W.l.o.g. B(R) = B := B(0, 1).

By the trace theorem, Lemma A.1, $f \in W^{s-\frac {n}{s},\frac {n}{s}}(\partial B)$. Let g be the harmonic extension of f to $\mathbb {R}^{n} \backslash B$. Then $g \in W^{s,\frac {n}{s}}(\mathbb {R}^{n} \backslash B)$, again by Lemma A.1. Also, since f is continuous on ∂B, g is also continuous. Set

$$ h := \begin{cases} g \quad &\text{in }\mathbb{R}^{n} \backslash B,\\ f \quad &\text{in }B.\\ \end{cases} $$

By Lemma A.3, $h \in W^{s,\frac {n}{s}}(\mathbb {R}^{n})$ and h is locally uniformly continuous on $\mathbb {R}^{n} \backslash B$. This last fact implies that

$$ h_{k}(x) := h\left ({\frac{k+1}{k}(x)} \right ) $$

converges uniformly to h on ∂B as $k \to \infty$, and also in $W^{s,\frac {n}{s}}_{loc}(\mathbb {R}^{n})$.

Now let us consider the standard mollification f_ε := h_k ∗ η_ε. For ε small enough in comparison with $\frac {1}{k}$, f_ε converges uniformly on ∂B to f and in $W^{s,\frac {n}{s}}(B))$. This completes the proof. □