Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

Abstract

A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are mostly of measure theoretic nature and reach beyond Lipschitz regular domains. Previous results were limited to regular geometric configurations or Hilbertian Sobolev spaces. Sets with porous boundary and their characteristic multipliers on smoothness spaces play a major role in the arguments.

Appendices

Appendix A: Porous Sets

We provide a streamlined approach to the geometry of porous sets. All this is known to the experts but some results require going through existing literature in a rather opaque way. The reader may look up relevant definitions in Sect. 2.1.

Lemma A.1

Every porous set $E\subseteq {\mathbb {R}}^{d}$ is a Lebesgue null set.

Proof

By Remark 2.7, each ball B centered in E contains a ball of comparable radius that does not intersect E. Hence, there is $\delta \in (0,1)$ depending only on E such that

$$\begin{aligned} \frac{|B\cap E|}{|B|} \le 1-\delta . \end{aligned}$$

By Lebesgue’s differentiation theorem this implies $\mathbb {1}_E = 0$ almost everywhere. $\square$

We recall the Vitali covering lemma that will be used frequently in the following, see [24, Theorem 1.2].

Lemma A.2

Let $\{B_i\}_{i\in I}$ be a family of open balls with uniformly bounded radii. Then there exists a subfamily $\{B_j\}_{j\in J}$ of disjoint balls such that

$$\begin{aligned} \bigcup _i B_i \subseteq \bigcup _j 5 B_j. \end{aligned}$$

Corollary A.3

Let $E\subseteq {\mathbb {R}}^d$ and $0<r\le R<\infty$. For any ball B of radius R the set $E\cap B$ can be covered by $10^d(R/r)^d$ ball of radius r centered in $E\cap B$.

Proof

Consider the covering $\{\mathrm {B}(x,r/5)\}_{x\in B\cap E}$ of $B \cap E$. We find a disjoint subfamily $\{B_i\}_{i\in I}$ such that $B\cap E \subseteq \cup _{i\in I} 5B_i$. We denote by $\#_i$ the cardinality of I and calculate

$$\begin{aligned} \#_i c_d (r/5)^d = |\cup _{i\in I} B_i| \le |2B| = c_d 2^d R^d, \end{aligned}$$

where $c_d$ is the measure of the unit ball. This shows $\#_i \le 10^d (R/r)^d$. $\square$

We continue with the simple observation that the radius bound by 1 in the definition of $\ell$-regularity is arbitrary.

Lemma A.4

Let $E\subseteq {\mathbb {R}}^d$ and $0<\ell \le d$. If for some $M \in (0,\infty )$ there is comparability ${\mathcal {H}}^\ell (B \cap E) \approx \mathrm {r}(B)^\ell$ uniformly for all open balls B of radius $\mathrm {r}(B) \le M$ centered in E, then the same is true for any $M \in (0,\infty )$.

Proof

Suppose we have uniform comparability for balls up to radius $\mathrm {r}(B) \le m$. Given $M > m$, we need to extend it to balls B centered in E of radius $\mathrm {r}(B) \le M$. Let $c {:=}m/M$. The calculation

$$\begin{aligned} \frac{m^\ell \mathrm {r}(B)^\ell }{M^\ell } \lesssim {\mathcal {H}}^l(cB \cap E) \le {\mathcal {H}}^\ell (B\cap E) \end{aligned}$$

gives the lower estimate. For the upper one, we cover $B\cap E$ by $10^d/c^d$ balls of radius $c \mathrm {r}(B)$ centered in $B \cap E$ according to Corollary A.3 and conclude ${\mathcal {H}}^\ell (B\cap E) \lesssim \mathrm {r}(B)^\ell$.

$\square$

We come to computing the Assouad dimensions of Ahlfors-regular sets.

Lemma A.5

Let $E\subseteq {\mathbb {R}}^d$ be $\ell$-regular for some $0<\ell \le d$ and let $M<\infty$. There exist constants $c, C> 0$ such that, if $x \in E$ and $0< r\le R < M$, then in order to cover $E \cap \mathrm {B}(x,R)$ by balls of radius r centered in E, at least $c(R/r)^\ell$ and at most $C(R/r)^\ell$ balls are needed. If E is unbounded and uniformly $\ell$-regular, then this also holds for $M = \infty$.

Proof

Let $\{B_i\}_{i\in I}$ be some cover of $E\cap \mathrm {B}(x,R)$ by balls of radius r. We use Lemma A.4 to calculate

$$\begin{aligned} R^\ell \lesssim {\mathcal {H}}^\ell (B(x,R) \cap E) \le {\mathcal {H}}^\ell (\cup _{i\in I} B_i \cap E) \le \sum _{i\in I} {\mathcal {H}}^\ell (B_i\cap E) \lesssim \#_i r^\ell , \end{aligned}$$

which shows $\#_i \gtrsim (R/r)^\ell$ and gives the constant c. As for C, we select a subfamily of disjoint balls $B_j$ from the covering $\{\mathrm {B}(x,r/5)\}_{x\in B\cap E}$ of $B\cap E$. Then we estimate, using Lemma A.4,

$$\begin{aligned} \#_j (r/5)^\ell \lesssim \sum _{j\in J} {\mathcal {H}}^\ell (B_j \cap E) \le {\mathcal {H}}^\ell (2B \cap E) \lesssim (2R)^\ell \end{aligned}$$

and conclude $\#_j \lesssim (R/r)^\ell$. $\square$

Proposition A.6

Let $E\subseteq {\mathbb {R}}^d$ be uniformly $\ell$-regular. It follows that ${\underline{\dim }}_{{\mathcal {A}}}{{\mathcal {S}}}(E)= {\overline{\dim }}_{{\mathcal {A}}}{{\mathcal {S}}}(E) =\ell$.

Proof

We can rephrase Lemma A.5 in the language of Definition 2.8. It precisely asserts that $\ell \in {\underline{{{\mathcal {A}}}{{\mathcal {S}}}}}(E) \cap {\overline{{{\mathcal {A}}}{{\mathcal {S}}}}}(E)$. Hence, we get ${\underline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}(E) \ge \ell$ and ${\overline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}(E) \le \ell$. The claim follows since ${\underline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}(E) \le {\overline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}(E)$ holds for any set E. Indeed, given $\lambda \in {\underline{{{\mathcal {A}}}{{\mathcal {S}}}}}(E)$ and $\mu \in {\overline{{{\mathcal {A}}}{{\mathcal {S}}}}}(E)$ we have $(R/r)^\lambda \lesssim (R/r)^\mu$ for all $0<r<R<{{\,\mathrm{diam}\,}}(E)$ and hence $\lambda \le \mu$. $\square$

We turn to porosity. The following result was already mentioned in Sect. 1.1.

Lemma A.7

Let $E \subseteq F\subseteq {\mathbb {R}}^d$. If F is $\ell$-regular and E is m-regular with $0<m<\ell \le d$, then E is porous in F. Likewise, if ${\overline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}(E) < {\underline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}(F)$, then E is uniformly porous in F.

Proof

We begin with the first claim. Lemma A.5 yields some $C\ge 1$ such that, if $x \in E$ and $0<r\le R\le 1$, then at most $C(2R/r)^m$ balls of radius r centered in E are needed to cover $E \cap \mathrm {B}(x,2R)$. It also yields some $c > 0$ such that at least $c(R/(2r))^\ell$ balls of radius 2r centered in F are needed to cover $F\cap \mathrm {B}(x,R)$. We use this observation with $r = \kappa R$, where $\kappa \in (0,1)$ satisfies $c/(2\kappa )^\ell > C(2/\kappa )^m$. This is possible due to $m < \ell$.

Let $\{B_i\}_{i \in I}$ be a family of $\#_i \le C(2/\kappa )^m$ balls of radius r centered in E that cover $E \cap \mathrm {B}(x,2R)$. By choice of $\kappa$ the balls $\{2B_i\}_{i \in I}$ cannot cover $F \cap \mathrm {B}(x,R)$. Pick $y \in F \cap \mathrm {B}(x,R)$ that is not contained in any of the $2B_i$. By construction we have $\mathrm {B}(y,r) \subseteq {\mathbb {R}}^d \setminus \cup _{i \in I} B_i$ but due to $r < R$ we also have $\mathrm {B}(y,r) \subseteq \mathrm {B}(x, 2R)$ and hence $E \cap \mathrm {B}(y,r) \subseteq \cup _{i} B_i$. Thus, we must have $E \cap \mathrm {B}(y,r) = \emptyset$ and conclude that E is porous in F.

The proof of the second claim is identical, but we do not assume $R \le 1$ and have the covering properties for some $m \in \overline{{{\mathcal {A}}}{{\mathcal {S}}}}(E)$ and $\ell \in {\underline{{{\mathcal {A}}}{{\mathcal {S}}}}}(F)$ with $m < \ell$ by assumption. $\square$

Lemma A.8

If $E\subseteq {\mathbb {R}}^d$ is porous, then there exist $C\ge 1$ and $0<s<d$ such that, given $x \in E$ and $0< r < R\le 1$, there is a covering of $E\cap B(x,R)$ by $C(R/r)^s$ balls of radius r centered in E. Moreover, if E is uniformly porous, then $\overline{\dim }_{{\mathcal {A}}}{{\mathcal {S}}}(E)<d$.

Proof

We only show the porous case since the uniform case again just follows by dropping all restrictions on the radii. In the following all cubes are closed and axis-aligned. As indicated in Sect. 2.1, we can equivalently replace balls by cubes and radii by side lengths in the definition of porosity and Assouad dimension. Likewise, it suffices to establish the claim of the lemma with cubes.

In view of Remark 2.7 we can fix $n\in {\mathbb {N}}$ such that for every cube $Q \subseteq {\mathbb {R}}^d$ there is a cube $Q' \subseteq Q \setminus E$ of sidelength $\ell (Q') = \ell (Q)/n$. We fix a cube Q centered in E of side length $R \le 1$. Let $0<r\le R$ and fix $k\in {\mathbb {N}}$ such that $R/(2n)^{k+1} \le r < R/(2n)^k$. We claim that we can cover Q by $((2n)^d-1)^{k+1}$ closed cubes of side length $R/(2n)^{k+1}$. Put $s{:=}\log ((2n)^d-1)/\log (2n)<d$. Then

$$\begin{aligned} ((2n)^d-1)^{k+1} = (2n)^s (2n)^{ks} < (2n)^s (R/r)^s \end{aligned}$$

shows the assertion.

For the claim we start with $k=1$. There is a cube $Q' \subseteq Q\setminus E$ of side length R / n. Then there is a cube $Q''$ in the grid of $(2n)^{d}$ cubes with sidelength R / (2n) covering Q that is contained in $Q'$. This means that we only need $(2n)^d-1$ cubes of side length R / (2n) to cover E. We conclude by applying this argument inductively on each cube of the previous covering. $\square$

Combining the uniform cases of the two preceding lemmas lets us re-obtain a result of Luukkainen [30, Thm 5.2]. Note that ${\underline{\dim }}_{{{\mathcal {A}}}{{\mathcal {S}}}}({\mathbb {R}}^d) = d$ due to Proposition A.6.

Proposition A.9

A set $E \subseteq {\mathbb {R}}^d$ is uniformly porous if and only if its upper Assouad dimension is strictly less than d.

We can use the non-uniform cases to show that some open sets are of class ${\mathcal {D}}^t$. The argument is a slight adaption of [29, Theorem 4.2].

Proposition A.10

Let $O\subseteq {\mathbb {R}}^d$ be open. If $\partial O$ is porous, then $O \in {\mathcal {D}}^t$ for some $t\in (0,1)$. If $\partial O$ is $\ell$-regular for some $0<\ell <d$, then $O \in {\mathcal {D}}^t$ for all $t \in (0, \max \{1, d -\ell \})$.

Proof

If $\partial O$ is porous, then we pick $C\ge 1$ and $0<s<d$ according to Lemma A.8 such that for each $j\ge 0$ and for any ball B with radius $r\le 1$ centered in $\partial O$ we can cover $B\cap \partial O$ by at most $C2^{js}$ balls of radius $r2^{-j}$. If $\partial O$ is $\ell$-regular, then Lemma A.5 guarantees that we can take $s = \ell$. In any case, fix $\max (s,d-1)<u<d$. Put $E_j{:=}\{x\in B: {{\,\mathrm{d}\,}}(x,\partial O) \le r2^{-j}\}$ and $A_j{:=}E_j \setminus E_{j+1}$. By construction, the covering property for $B\cap \partial O$ implies that we can cover $E_j$ by at most $C2^{js}$ balls of radius $r2^{-(j-1)}$. The d-regularity of Lebesgue measure then implies

$$\begin{aligned} |A_j| \le |E_j| \lesssim 2^{js} r^d 2^{d-jd}. \end{aligned}$$

(41)

We use that $\{A_j\}_{j\ge 0}$ is a disjoint cover of $B \setminus \partial O$, comparability ${{\,\mathrm{d}\,}}(x,\partial O) \approx r2^{-j}$ on $A_j$, estimate (41), and $s<u$ to calculate

$$\begin{aligned} \int _{B\setminus \partial O} {{\,\mathrm{d}\,}}(y,\partial E)^{u-d} \mathrm {d}y&\le \sum _j \int _{A_j} {{\,\mathrm{d}\,}}(y,\partial O)^{u-d} \mathrm {d}y \lesssim \sum _j |A_j| 2^{dj-u j} r^{u-d}\nonumber \\&\lesssim \sum _j r^u 2^{j(s-u)} \lesssim r^u. \end{aligned}$$

Setting $t {:=}d-u\in (0,1)$, we write this in the form

$$\begin{aligned} \sup _{x\in \partial O} \sup _{0<r\le 1} r^{t-d} \int _{\mathrm {B}(x,r)\setminus \partial O} {{\,\mathrm{d}\,}}(y,\partial O)^{-t} < \infty , \end{aligned}$$

which just means that $O \in {\mathcal {D}}^t$. In the case of $\ell$-regular boundary, every $u \in (\max \{\ell ,d-1\},1)$ and thus every $t \in (0,\max \{1, d-\ell \})$ was admissible in the proof. $\square$

Appendix B: Intrinsic Characterizations

Although perfectly suited for interpolation questions, the function spaces $\mathrm {X}^{s,p}(O) = \mathrm {X}^{s,p}({\mathbb {R}}^d)|_O$ lack an intrinsic characterization through a norm that only uses information on O. The problem of finding such characterizations has a long history and we refer for instance to [15, 21, 27, 34, 36, 40, 41, 45] and references therein. Here, we only mention two results that are of particular importance for putting our paper into context of work on mixed boundary value problems [3, 8, 11, 13, 22, 23, 39].

The following is the full-dimensional case in [27, Theorem VI.1].

Proposition B.1

Let $O \subseteq {\mathbb {R}}^d$ be an open d-regular set and let $p \in (1,\infty )$ and $s \in (0,1)$. Then $\mathrm {W}^{s,p}(O)$ is up to equivalent norms the space of those $f \in \mathrm {L}^p(O)$ for which

$$\begin{aligned} \Vert f\Vert {:=}\Vert f\Vert _{\mathrm {L}^p(O)} + \bigg (\iint _{O \times O} \frac{|f(x) - f(y)|^p}{|x-y|^{d+ sp}} \; \mathrm {d}x \; \mathrm {d}y \bigg )^{1/p} < \infty . \end{aligned}$$

Remark B.2

If in the setting above $D \subseteq \overline{O}$ is $(d-1)$-regular, then $\mathrm {W}^{s,p}_D(O)$ is the closure of $\mathrm {C}_D^\infty (O)$ for the intrinsic norm $\Vert \cdot \Vert$. This follows from Lemma 3.3.

Probably most important result of the above type concerns $\mathrm {W}^{1,p}_D(O)$.

Proposition B.3

Suppose $O \subseteq {\mathbb {R}}^d$ is an open set, $D \subseteq \overline{O}$ is $(d-1)$-regular, and O satisfies a uniform Lipschitz condition around $\overline{\partial O \setminus D}$. Then $\mathrm {W}^{1,p}_D(O)$ can equivalently be normed by

$$\begin{aligned} \Vert f\Vert {:=}\Big ( \Vert f\Vert _{\mathrm {L}^p(O)}^p + \Vert \nabla f\Vert _{\mathrm {L}^p(O)}^p \Big )^{1/p} \end{aligned}$$

and in fact it is the closure of $\mathrm {C}_D^\infty (O)$ for this norm.

Proof

Let X be the closure of $\mathrm {C}_D^\infty (O)$ for the norm $\Vert \cdot \Vert$. Clearly we have $\mathrm {W}^{1,p}_D(O) \subseteq X$ with continuous inclusion. For the converse inclusion we argue as in [3, Lemma 3.2], using the localization formalism of Sect. 4.4 and in particular the maps ${\mathcal {E}}$ and ${\mathcal {R}}$ defined in (14) and (15). We let $\sigma$ be the extension of functions from $(0,1) \times (-1,1)^{d-1}$ to $(-1,1)^{d}$ by even reflection and consider the operator

$$\begin{aligned} {\mathcal {T}}: f \mapsto {\mathcal {R}}(({\mathcal {E}}f)_0, (\sigma ({\mathcal {E}}f)_i)_{i \in I}). \end{aligned}$$

By construction ${\mathcal {T}}f|_O = f$ holds for $f \in X$. It follows from the proof of Lemma 4.10 that ${\mathcal {T}}: X \rightarrow \mathrm {W}^{1,p}({\mathbb {R}}^d)$ is bounded. The proof of Lemma 4.12 reveals that ${\mathcal {T}}$ in fact maps X into $\mathrm {W}^{1,p}_D({\mathbb {R}}^d)$, which implies $X \subseteq \mathrm {W}^{1,p}_D(O)$. $\square$