Strong non-regularity of the prescribed Jacobian equation near $L^\infty$ and consequences

Speaker(s)

Jakub Takáč

Language of the talk

English

Date

March 11, 2026, 12:30 p.m.

Room

room 4060

Seminar

Seminarium Zakładu Równań i Analizy

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It has long been known that for underdetermined differential equations of the form
$$
    \mathcal{A} u = f,
$$
where $\mathcal{A}$ is some underdetermined differential operator of order $1$, one cannot expect to find a Lipschitz solution $u$ for every right side $f\in L^\infty$. The general statement (for every such linear operator) is known only colloquially, but in some specific cases, proofs are available (e.g.~when $\mathcal{A}=\textnormal{div}$ due to Preiss and independently McMullen).

When $\mathcal{A}=\det \textnormal{D}$ (where $D$ is the Frech\'et differential and $u$ is a vector field), in which case the operator is not linear, some information in this direction is also known. Most crucially, due to Burago and Kleiner, we know that there exist $L^\infty$ right sides $f$ for which there is no \emph{biLipschitz} solution.

In my recent work, motivated by resolving the Flat Chain Conjecture concerning the structure of metric currents, I show the following stronger result. Consider the subset of $L^\infty$ consisting of the functions $f$ which may be written as
$$
    f= \det\textnormal{D}u \quad\textnormal{for some $1$-Lipschitz $u$.}
$$

Then the weak$^*$ closure of the convex hull of this subset is weak$^*$ nowhere dense. In particular, for any $\varepsilon>0$, one may find a function $f\in L^\infty$, $|| f ||_\infty <\epsilon$ and a weak$^*$ open neighbourhood $U$ of $f$ such that \emph{no function $g\in U$} may be weak$^*$ approximated by convex combinations of $1$-Lipschitz Jacobians.

From the perspective of PDEs, likely the most interesting progress is in improving the result of Burago and Kleiner from biLipschits mappings to Lipschitz mappings. However, for the application in the theory of metric currents, working with the convex combinations as well as the weak$^*$ topology is also crucial.

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