Constraint Maps: Optimal conditions for $C^{1,1}$-regularity
OpenYear of origin: 2025
Posted online: 2025-07-13 07:39:47Z by Henrik Shahgholian2
Cite as: P-250713.1
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
What is the optimal regularity on $M$ which ensures that a minimizing constraint map $u\in W^{1,2}(\Omega;\overline M)$ is in $C^{1,1}_{loc}(\Omega\setminus \Sigma(u))$?
In [5], [3] we proved $C^{1,1}$-regularity of $u$ around regular points under the assumption that $\partial M\in C^{3,\alpha}$ for some $\alpha>0$, while $W^{2,p}$-regularity for $p< \infty$ only requires that $\partial M \in C^2$. This Problem asks whether this gap can be closed. A reasonable guess is that perhaps the optimal regularity of $u$ holds even for obstacles of class $\partial M\in C^{1,1}$.
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Created at: 2025-07-13 07:39:47Z
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