Quantitative Rigidity for Aviles–Giga Differential Inclusions

Definitions and mathematical statement

The supplied record does not define the intended Aviles--Giga set $K$. Consequently, the following is a conditional transcription of the supplied question, pending specification of $K$, rather than a complete formulation of the intended problem.

For the intended planar interpretation, let $B_r=\{x\in{\mathbb{R}}^2:|x|<r\}$. Let $K\subseteq{\mathbb{R}}^{2\times2}$ denote the particular nonempty matrix set intended by the source; its elements and defining parametrization are not supplied. For a matrix $M\in{\mathbb{R}}^{2\times2}$, use the Frobenius norm and define \[ \operatorname{dist}(M,K)=\inf_{A\in K}|M-A|. \] The Sobolev space $H^1(B_1;{\mathbb{R}}^2)$ consists of square-integrable vector fields whose first weak derivatives are square-integrable, and $Dw$ denotes the weak Jacobian matrix of $w$.

For this fixed intended set $K$, do there exist constants $C>0$ and $\alpha\in(0,1)$, independent of $w$, such that \[ \forall w\in H^1(B_1;{\mathbb{R}}^2),\qquad \inf_{A\in K}\int_{B_{1/2}}|Dw(x)-A|^2\,{\,\mathrm{d}} x \leq C\left(\int_{B_1}\operatorname{dist}(Dw(x),K)^2\,{\,\mathrm{d}} x\right)^\alpha? \] Here the matrix $A$ on the left is constant in space. The record supplies no bound, normalization, or smallness assumption on $w$, $Dw$, or the right-hand integral; none has been inserted. Whether an additional hypothesis is intended requires review.

The surrounding source excerpt describes $K$ through an unspecified map $P$: an exact inclusion $Dw\in K$ almost everywhere is said to admit a representation $Dw=P(m)$ for a field $m:\Omega\to S^1$, where $S^1=\{v\in{\mathbb{R}}^2:|v|=1\}$. Neither $P$ nor a formula for $K$ is provided. The excerpt explicitly says that $\operatorname{div}m=0$ in $\mathcal D'(\Omega)$ is not an a priori assumption. Its discussion of rigidity in the sense of Jabin--Otto--Perthame does not itself specify the missing set or additional hypotheses for the displayed estimate.

Short English statement

The question asks whether the squared $L^2$ distance of $Dw$ from one constant matrix in the intended Aviles--Giga set $K$, measured on $B_{1/2}$, is controlled by a fixed sublinear power of its squared pointwise distance from $K$ on $B_1$. The definition of $K$ is missing, and the unrestricted quantifier over $H^1$ requires review. This is a statement-only draft.

Sources

• Local UnsolvedMath registry, permanent ID $30005210$, title: "Quantitative Rigidity for Aviles--Giga Differential Inclusions." Exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2022; no proposer is identified.
• Supplied source citation: Calculus of Variations (2023), Oberwolfach Reports, pp. 2129--2193, DOI 10.4171/owr/2022/37. Original report: https://doi.org/10.4171/owr/2022/37. These bibliographic details are reproduced as supplied.
• Supplied literature reference: Lamy--Lorent--Peng, "quantitative rigidity," https://doi.org/10.1093/imrn/rnad108.
• Supplied literature reference: Lamy--Lorent--Peng, "non-elliptic curves," https://arxiv.org/abs/2404.02121.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that quantitative rigidity is proved for elliptic differential-inclusion curves and exact rigidity is known for some non-elliptic curves, while the stated approximate estimate for the Aviles--Giga set $K$ remains unresolved. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked and its notation repaired. These are supplied verification claims; no independent source consultation or mathematical verification is asserted here. The surrounding excerpt mentions Jabin--Otto--Perthame rigidity and numbered references [2], [4], and [5], whose bibliographic identifications are not supplied.