Open problems in Mathematics stemming from research papers

Constraint maps into a spherical cap

This problem pertains to the more geometric setting of constraint maps between Riemannian manifolds. ...
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian81

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- Analysis of PDEs
Minimality of the vortex

Is it the case that the radial map considered in Proposition 3.1, in [2] and in [5] is a minimizing constraint map in low dimensions, specifically $3\leq n\leq 6$? ...
SolvedYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian66

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- Analysis of PDEs
Singular FB produced by convex obstacles

In [5] we showed that for a large class of convex obstacles, and for a large class of boundary data, there are necessarily singularities of the corresponding minimizing constraint maps on their free boundaries. Is it the case that, at least in some of these examples, also the free boundaries are singular? ...
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian42

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- Analysis of PDEs
Fine structure of FB

Consider the free boundary of a constraint map $u$ in the neighborhood of a continuity point $x_0\not\in \Sigma(u)$. Suppose also that $|Du(x_0)|>0$. Does the fine structure analysis of the free boundary from the scalar obstacle problem [6] carry through to the free boundary of $u$ near $x_0$? ...
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian43

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- Analysis of PDEs
Stable constraint maps

For suitable targets, such as when $M=\Bbb R^m\setminus \mathbb B^m$ for $m\geq 4$, is there a suitable notion of stability for constraint maps so that properties (i)--(iv) of Definition \ref{def:F} hold for such maps? ...
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian34

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- Analysis of PDEs
UCP and low rank

Let $M^c$ be convex and $u\in W^{1,2}(\Omega;\overline M)$ be a minimizing constraint map. If $\text{rank}(Du)\leq 1$ in an open subset of $\Omega$, do we have $\text{rank}(Du)\leq 1$ in $\Omega$? ...
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian75

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- Analysis of PDEs
Constant $\Pi$ on the boundary

Let $u\in W^{1,2}(B;\overline M)$ be a $C^{1,1}$ weakly constraint map. If $\Pi \circ u$ is constant on $\partial B$, is it the case that $\Pi\circ u$ is constant in $B$, at least when $M^c$ is convex? What are the optimal geometric conditions on $M$ for this to hold? ...
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian40

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- Analysis of PDEs

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