A remark on staircase laminates in restricted sets

In [5], Kleiner, Müller, Székelyhidi, and Xie provided a very useful framework for convex integration via laminates, introducing the notion of L p -reducibility between sets. As applications, they provided streamlined proofs for several results, including the counterexample of Colombo-Tione [3] to the conjecture of Iwaniec-Sbordone [4].

Notably missing and explicitly excluded from these examples, cf. [5, p.1574], is the following classical result by Astala-Faraco-Székelyhidi [2]: Whenever A : B m → R m×m is a symmetric, bounded, measurable, uniformly elliptic matrix, i.e., 1 Λ I ≤ A ≤ ΛI a.e. in B m , then Gehring’s lemma provides Meyers improved regularity result: a solution u ∈ W 1,2 0 (B m ) to div(A∇u) = 0 in B m satisfies ∇u ∈ L 2+ε (B m ). The precise size of ε (in dependence on the ellipticity constant) is unknown in general. The following result provides an upper bound for ε, which is sharp in two dimensions, see [1,6,7]. In this note, we extend the framework of [5] to include Theorem 1.1.

To this end, we introduce, for open sets U ⊂ R d×m , the notion of laminate with construction steps in U (see Definition 3.3) and the notion of U -reduced in weak L p (see Definition 2.3).

The following result says that if U can be U -reduced to K in weak L p then we can find many exact solutions to the differential inclusion ∇w ∈ K. This is a slight extension of [5,Theorem 4.1], which is Theorem 1.2 for U = R d×n .

Theorem 1.2 (Exact solution, cf. [5,Theorem 4.1]). Let K ⊂ R d×m , U ⊂ R d×m open, and 1 < p < ∞ such that U can be U -reduced to K in weak L p , cf. Definition 2.3.

Then for any regular domain Ω ⊂ R m , any X 0 ∈ U , b ∈ R d and any δ > 0, α ∈ (0, 1) there exists a piecewise affine map w ∈ W 1,1 (Ω, R d ) ∩ C α ( Ω, R d ) with w(x) = l X 0 ,b (x) ≡ X 0 x + b on ∂Ω such that ∇w(x) ∈ K a.e. x ∈ Ω ∥w -l X 0 ,b ∥ C α ( Ω,R d ) < δ and there exists a constant C > 0 such that for all t > 0, (1.1) {x ∈ Ω : |∇w(x)| > t} ≤ Ct -p .

If moreover, U can be U -reduced to K exactly in weak L p then for another constant c > 0,

In order to obtain the assumptions of Theorem 1.2 we have a laminate criterion, which is a slight adaptation of [5,Theorem 4.3].

Theorem 1.3 (Staircase laminate criterion, cf. [5,Theorem 4.3]). Let K ⊂ R d×m , U ⊂ R d×m open, and 1 < p < ∞. Assume that there exist constants M ≥ 1 and m > 0 with the following property: for any X 0 ∈ U there exists a staircase laminate ν ∞ X 0 supported on K, with barycenter X 0 and construction steps in U , see Definition 3.5, satisfying the bound

for all t > 0.

Then U can be U -reduced to K in weak L p .

If, in addition to (1.3), we also have

then U can be U -reduced to K exactly in weak L p .

Outline. In Section 2 we introduce the notion of U -reducibility. We then adapt the proofs in [5] to establish Theorem 1.2. In Section 3 we introduce the notion of construction steps in U and prove Theorem 1.3. Finally, in Section 4 we apply this framework to obtain a streamlined proof of Theorem 1.1 using L p -reducibility.

Notation. We use the standard notation ≾, ≍, and ≿. We write A ≾ B if there exists a constant C > 0, depending only on irrelevant data, such that

We write A ≍ B if A ≾ B and B ≾ A.

Acknowledgment. The project is co-financed by Armin Schikorra appreciates the hospitality of the Thematic Research Programme Geometric Analysis: Methods and Applications at the University of Warsaw, where part of this work was conducted. Armin Schikorra and Pholphum Kamthorntaksina’s research was supported in part by the International Centre for Theoretical Sciences (ICTS) for participating in the program -Geometric Analysis and PDE 2026 (code: ICTS/GPDE2026/02). Armin Schikorra was Alexander von Humboldt Research Fellow.

We begin by defining the U -reducibility, which essentially just means that the laminate construction process can restart from within U . We also recall the following two standard definitions.

is open, bounded, connected, and ∂Ω has zero m-dimensional Lebesgue measure.

Let Ω ⊂ R m be a regular domain. We call a map w ∈ W 1,1 (Ω, R d ) piecewise affine if there exists an at most countable decomposition Ω = i Ω i ∪ N into pairwise disjoint regular domains Ω i ⊂ Ω and a null set N such that w agrees with an affine map on each Ω i . That is, for any i there exists X i ∈ R d×m and b i ∈ R d such that w(x) = X i x + b i for all x ∈ Ω i . We also denote Ω = i Ω i the open subset of Ω where u is locally affine. Note that regular domains Ω i are exactly the connected components of Ω.

The main difference of the following notion of reducibility in U to the original version in [5] is the assumption (2.2). Definition 2.3 (Reducibility and exact reducibility in weak L p ). For U, K ⊂ R d×m and 1 < p < ∞ we say that U can be U -reduced to K in weak L p provided there exists a constant M = M (U, K, p) ≥ 1 satisfying for an arbitrary

, and a regular domain Ω ⊂ R m the following property:

There exists a piecewise affine map

and on the error set

(2.

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