Intermediate Regularity Regime for Monge–Ampère Solutions

Definitions and mathematical statement

The supplied question concerns the intermediate regularity range \[ \frac13<\alpha<\frac23 \] for the equation \[ \operatorname{Det}\nabla^2v =-\frac12\operatorname{curl}\operatorname{curl}(\nabla v\otimes\nabla v)=f. \] The source does not specify the domain, the function space indexed by $\alpha$, the assumptions on $f$, boundary conditions, or the meaning of the separate term "weak solution." Consequently, the following definitions explain the displayed operator under a planar interpretation; they do not supply missing hypotheses for the original question.

For this interpretation, let $\Omega\subset{\mathbb{R}}^2$ be open. For a matrix field $A=(A_{ij})_{i,j=1}^2$, set \[ \operatorname{curl}\operatorname{curl}A :=\partial_{22}A_{11}+\partial_{11}A_{22} -\partial_{12}A_{12}-\partial_{21}A_{21}, \] with derivatives understood in the sense of distributions. If $v\in W^{1,2}_{\mathrm{loc}}(\Omega)$, then $(\nabla v\otimes\nabla v)_{ij}=\partial_i v\,\partial_j v$ is locally integrable, so the displayed very weak Hessian determinant is well defined as a distribution. Writing $v_i=\partial_i v$, the equation with $f\in\mathcal D'(\Omega)$ means \[ -\frac12\int_\Omega \left(v_1^2\partial_{22}\varphi +v_2^2\partial_{11}\varphi -2v_1v_2\partial_{12}\varphi\right)\,{\,\mathrm{d}} x =\langle f,\varphi\rangle \quad\text{for every }\varphi\in C_c^\infty(\Omega). \] For smooth $v$, this operator agrees with $\det\nabla^2v$.

If the intended regularity is $v\in C^{1,\alpha}_{\mathrm{loc}}(\Omega)$, this means that $v$ is continuously differentiable and, for every compact $K\subset\Omega$, there is $C_K<\infty$ such that \[ |\nabla v(x)-\nabla v(y)|\le C_K|x-y|^\alpha \quad\text{for all }x,y\in K. \] This interpretation of $\alpha$ is not explicitly specified in the supplied record and requires confirmation.

Source question. Determine the qualitative behavior of weak and very weak solutions of the displayed Monge--Ampère equation in the regime $1/3<\alpha<2/3$. The source does not identify a particular rigidity, convexity, flexibility, existence, or uniqueness assertion to prove, nor does it specify the quantifiers over domains and right-hand sides. Thus this is a research question with unresolved formulation details, rather than a fully specified conjecture.

The supplied dated assessment describes rigidity and convexity above the two-thirds threshold and flexible constructions below the one-third threshold, while describing the intermediate range as unresolved. These are supplied literature qualifications, not independently verified mathematical assertions; their applicable hypotheses are absent from the record. The endpoints $\alpha=1/3$ and $\alpha=2/3$ are excluded from the stated question.

Short English statement

The question asks how weak and very weak Monge--Ampère solutions behave when $1/3<\alpha<2/3$, between the rigidity and flexibility regimes described by the supplied literature assessment. A precise conjecture still requires the intended regularity space, domain, assumptions on $f$, solution notions, boundary conditions, and a specific qualitative conclusion. The supplied status is "partially solved"; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002988$, source problem code catalogue entry, title "Intermediate Regularity Regime for Monge--Ampère Solutions," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report attribution: Mini-Workshop: Scales in Plasticity (2016), Oberwolfach Reports, pp. 3007--3048, DOI 10.4171/owr/2015/53. https://doi.org/10.4171/owr/2015/53. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report citation is supplied provenance; its contents and the precise location of this question have not been independently verified.
• Supplied literature source, "Recent very weak Monge--Ampere results": https://link.springer.com/article/10.1007/s00526-025-03019-0.
• Supplied literature source, "Intermediate-regularity analysis": https://arxiv.org/abs/2310.06693.
• Supplied literature source, "Convex-integration constructions": https://sites.pitt.edu/~lewicka/preprints/Flexible-Monge9.pdf.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the problem "partially solved," describes rigidity and convexity above the two-thirds threshold and flexible constructions below the one-third threshold, and describes the intermediate range as unresolved. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified, with the explanation that notation and publication prose were repaired without changing the question's substance. This is a supplied editorial verification claim, not mathematical review of this draft or independent source verification.