Singular FB produced by convex obstacles
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian
Cite as: P-250713.7
General Description View the group
We propose several open problems that likely represent just the tip of the iceberg in the emerging theory of Constraint Maps, a field recently revitalized by the current authors. Before exploring the problems listed below, we recommend that interested readers consult the two main survey papers [1], [2], which provide a more accessible introduction to the topic.
The mathematical problem of interest is to study minimizers (or critical points of) $u: \Omega \to \overline{M}$ of the Dirichlet energy (in an appropriate space) given by \begin{equation} E(u) = \frac{1}{2} \int_{\Omega } \left( | D u|^2 + G(x,u) \right) \, dx. %\qquad (\Omega \subset N_1, \quad dx= dvol_{N_1}). \end{equation} Here $D u $ is the gradient of the vector-valued function $u$, $G(x,u)$ is a given function, and if necessary appropriate boundary values are prescribed. Here we also have $\Omega \subset \Bbb R^n$, and $M \subset \Bbb R^m$. In general $M^c$ being convex is a central assumption in order to obtain so-called free boundary.
In general one can consider maps $u$ from one manifold to another $u: N_1 \to N_2$, where $\Omega \subset N_1 $. Since most if not all questions we consider are of local nature, boundary values are ignored. The class of maps $u$ are also subject to the constraint, i.e., the image of the maps remains within (the closure of) a prescribed domain $M \subset N_2$, with boundary.
Since the case where $N_2 = \mathbb{R}$ and $M = \mathbb{R}_+$ corresponds to the scalar setting commonly known as the obstacle problem and is well understood, we will throughout this proposal assume that $N_2$ has dimension greater than one.
The local nature of the problem also suggests that maps from an $n$-dimensional manifold $N_1$ may, without loss of generality, be treated as maps from $\mathbb{R}^n$. In this setting, the governing equations are replaced by their counterparts involving the Laplace–Beltrami operator (the so-called tension field). This substitution poses primarily technical challenges, rather than introducing fundamental conceptual difficulties, and we thus assume $N_1 = \Bbb R^n$. In our problems we shall also for the sake of presentation, and to avoid technical challenges, we only consider flat target manifolds, i.e., $N_2 = \Bbb R^m$.
We recall a few definitions from the references [1], [2]:
$u^{-1}(M)$: Non-coincidence set, the bulk for Euler equation through outer variation.
$u^{-1}(\partial M)$: Coincidence set, the bulk for Euler equation through inner variation.
$\mathcal{F}(u):= \partial u^{-1}(M)$: Free boundary, the unknown boundary between the above regions.
$\Sigma(u)$: The set of discontinuities of the map.
We also need to present the distance and projection components of the maps, by decomposing the map $u$ into two components \begin{equation} u= (\rho \circ u) \nu\circ u + \Pi \circ u, \end{equation} where $\rho (u (x)) = \hbox{dist} (x, M^c)$, and $\Pi$ projection onto $M^c$, and $\nu$ the unit normal on $\partial M$ pointing into $M$.
Below, we present a collection of open problems, grouped together for convenience but formulated as distinct, individual challenges. These problems can be found at [2].
Problem's Description
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?
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Created at: 2025-07-13 07:39:47Z
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