Fine-scale regularity of constraint maps with free boundaries in the obstacle case: the geometric approximation property in dimensions $n\ge3$, optimal regularity of the image map, vectorial fine structure, analyticity, and leaky maps

Constraint maps -- Dirichlet-energy minimizers $u:\Omega\to\overline M$ confined to the closure of a smooth set $M\subset\mathbb R^m$ with compact complement -- are the vectorial analogue of the classical obstacle problem: the interface $\partial u^{-1}(M)$ between the non-coincidence set $u^{-1}(M)$ and the coincidence set $u^{-1}(\partial M)$ is a free boundary, and the Euler-Lagrange system $\Delta u=A(Du,Du)\chi_{u^{-1}(\partial M)}$ couples the distance part $\rho\circ u$, a scalar obstacle problem, to the projected image map $\Pi\circ u$. Figalli, Kim and Shahgholian (2024) proved that constraint maps are $C^{1,1}$, that the free boundary is $C^\infty$ near regular points, that $D^3(\Pi\circ u)$ lies in BMO in all dimensions and is bounded near regular points, and that the optimal $C^{2,1}$ bound for the image map holds in the plane -- the last conditional on a geometric approximation property (GAP) for the scalar obstacle problem, known in two dimensions only through unpublished work of Eberle and Serra. The paper records five interlocking open questions, which state the problem: the GAP in dimensions $n\ge3$ and the conjecture that it holds near singular sets without anomalous points; optimal $C^{2,1}$ regularity of the image map in $n\ge3$ and whether it can be proved without the GAP; the fine structure of the singular free boundary and generic regularity in the vectorial setting; analyticity of the free boundary near regular points; and a complete two-dimensional theory for non-variational "leaky" solutions. All five components are currently open.

Background

In its normalized form, the classical obstacle problem asks for nonnegative solutions $w$ of

$$\Delta w=g\,\chi_{\{w>0\}},$$

in a bounded domain $\Omega\subset\mathbb R^n$, where $\chi_{\{w>0\}}$ is the characteristic function of the positivity set and $g\ge0$ is a given right-hand side. The interface $\partial\{w>0\}$ between the contact set $\{w=0\}$ and the positivity set is not prescribed by the data; it is a free boundary. Caffarelli's regularity theory for this problem splits the free boundary into regular points -- where the contact set has positive density, the free boundary is locally a smooth graph, and $w$ inherits the smoothness of the data -- and singular points, where the contact set has zero density and $w$ is well approximated by quadratic polynomials. The fine geometry of the singular set is subtler: Figalli and Serra proved that for $n\ge3$ the singular set has Hausdorff dimension at most $n-3$ (Hausdorff dimension measures the size of a set by how many balls of radius $\delta$ are needed to cover it as $\delta\to0$, and agrees with the usual dimension on smooth surfaces) and is rectifiable -- covered, up to a negligible set, by countably many smooth $(n-3)$-dimensional surfaces -- away from a smaller exceptional set of anomalous points, around which no fine structure is known; Figalli, Ros-Oton and Serra proved that for generic data the free boundary is smooth; Franceschini and Zatoń refined the partial regularity of the singular set itself; and Eberle, Figalli and Weiss completed the classification of global solutions -- the coincidence set of a global solution is a half-space, an ellipsoid, a paraboloid, or a cylinder with ellipsoid or paraboloid base -- after earlier work on global solutions by Eberle, Shahgholian and Weiss.

A constraint map is the vectorial counterpart of the obstacle problem. Let $\Omega\subset\mathbb R^n$ be a bounded domain and $M\subset\mathbb R^m$ ($m\ge2$) a smooth open set whose complement is compact and connected; $M^c$ plays the role of the obstacle. A constraint map is a minimizer (or local minimizer) of the Dirichlet energy $\int_\Omega|Dv|^2dx$ in the Sobolev class $W^{1,2}(\Omega;\overline M)$ of square-integrable maps with square-integrable first derivatives and values in the closure $\overline M$, subject to fixed boundary data. It satisfies the Euler-Lagrange system

$$\Delta u=A(Du,Du)\,\chi_{u^{-1}(\partial M)}$$

in the distributional sense, where $A$ is the second fundamental form of the hypersurface $\partial M$ evaluated on the row vectors of the differential $Du$ (a quadratic term measuring how the target boundary curves), and $u^{-1}(\partial M)$ is the coincidence set, where $u$ touches the obstacle. The interface between the non-coincidence set $u^{-1}(M)$, where the constraint is inactive, and the coincidence set is again a free boundary; its points are classified as regular or singular through the density of the coincidence set, exactly as in the scalar problem, because the free boundary of $u$ is fully characterized by that of the scalar function $\rho\circ u$ defined next.

Near $\partial M$ the map decomposes in tubular coordinates, in which each point near $\partial M$ is described by its nearest point on $\partial M$ and its signed distance to it: if $\rho$ is the signed distance to $\partial M$ (positive inside $M$) and $\Pi$ the nearest-point projection onto $\partial M$, then $u=\Pi\circ u+\rho\circ u\,\nu_{\Pi\circ u}$, with $\nu$ the inward unit normal. The distance part $w:=\rho\circ u$ is nonnegative and solves a scalar obstacle problem of the above form, with a right-hand side $g$ determined by the curvature of $\partial M$ and $Du$; the free boundary of $u$ is the free boundary of $w$. The genuinely new object is the image map $V:=\Pi\circ u$: modulo lower-order curvature terms it solves the coupled system

$$\Delta v=a\cdot Dw,\qquad\Delta w=g\,\chi_{\{w>0\}},\quad w\ge0,$$

with coefficients $a,g\in C^{1,\alpha}$. Since $w\in C^{1,1}$ (the optimal scalar regularity), elliptic regularity gives $D^3v\in BMO$ but not $D^3v\in L^\infty$; extracting the optimal $C^{2,1}$ estimate for $v$ requires finer properties of $w$ beyond its $C^{1,1}$ regularity, and this is where the open problems live. Here and below $C^{k,\alpha}$ is the Hölder class of functions whose derivatives of order $k$ are $\alpha$-Hölder continuous, and BMO is the space of locally integrable functions with bounded mean oscillation.

Figalli, Kim and Shahgholian established the foundational theory of the obstacle case. Every local constraint map satisfies $u\in C^{1,1}_{loc}(\Omega\setminus S)$ for a closed set $S$ of Hausdorff dimension at most $n-3$ (Theorem 2.1). Around every regular free-boundary point, the free boundary is a $C^\infty$ graph and $u$ is $C^\infty$ up to it from both sides (Theorem 2.2) -- but the proof, which uses a partial hodograph-Legendre transformation (an invertible change of variables that exchanges a spatial coordinate with a derivative of the solution, thereby straightening the free boundary into a fixed hyperplane) and regularity theory for elliptic transmission problems (elliptic equations whose coefficients jump across an interface, here the free boundary), does not yield analyticity. For the image map: if $u$ is continuous in a ball, then $D^3(\Pi\circ u)\in BMO_{loc}$ there, and $D^3(\Pi\circ u)\in L^\infty_{loc}$ if all free-boundary points in the ball are regular (Theorem 2.4); at a singular point where the distance part is pointwise $C^{2,\bar\sigma}$, the image map is pointwise $C^{3,\sigma}$ (Theorem 2.5), and the fine-structure results suggest this holds on the whole singular part up to a set of Hausdorff dimension $n-3$ (the exceptional set of the scalar theory, which exists for $n\ge3$). In the plane the optimal result holds in full: $D^3(\Pi\circ u)\in L^\infty_{loc}(\Omega)$ (Corollary 2.7). The $C^{2,1}$ threshold is optimal: the paper constructs a constraint map whose image map lies in $C^{2,1}$ but not in $C^3$ (Example 2.10), no matter how smooth the target.

The two-dimensional theorem is reduced, through Theorem 2.6, to a new quantitative property of the scalar obstacle problem isolated by the authors: the geometric approximation property (GAP, Definition 2.8). A nonnegative solution $w$ of $\Delta w=g\chi_{\{w>0\}}$ admits the GAP at a free-boundary point $x_0$, with order $2+\sigma$, if for some constant $\lambda>0$ there holds

$$\|w-U^r_{x_0}\|^*_{C^1(B_r(x_0)\cap\Omega)}\le\lambda\,\max\{r^3,\,g(x_0)^{1-\sigma}r^{2+\sigma}\}$$

for every $0<r<\operatorname{dist}(x_0,\partial\Omega)$, where $U^r_{x_0}$ is a nonnegative global solution of $\Delta U=g(x_0)\chi_{\{U>0\}}$ in all of $\mathbb R^n$ with $U(x_0)=0$, allowed to vary with $r$, and $\|f\|^*_{C^1(B_r)}:=\sup_{B_r}|f|+r\sup_{B_r}|Df|$ is the adimensional $C^1$-norm. The GAP is thus a quantitative local-to-global principle: at every scale, the local solution is $C^1$-close to an explicit global model, uniformly in the scale. It holds automatically at scales $r\ge g(x_0)$ when $g$ is Lipschitz, and in the plane it holds at all scales by a recent result of Eberle and Serra that the published paper cites as work in progress; in $n\ge3$ its validity is unclear, precisely because of the anomalous points of the scalar theory. The planar proof of the GAP rests on Sakai's explicit representation of planar global solutions via the Schwarz function -- the holomorphic function whose real part recovers the defining function of an analytic curve -- together with the classification of planar global solutions and explicit Newtonian potential estimates (the Newtonian potential is the function obtained by convolving a charge distribution with the fundamental solution of $\Delta$).

Finally, the paper observes that most of its arguments use only the Euler-Lagrange system, not the minimizing property, and it extends part of the theory to leaky maps: distributional solutions of the same system that need not satisfy the one-sided constraint $u(\Omega)\subset\overline M$, so that the distance part $\rho\circ u$ may change sign. Leaky maps are the vectorial analogue of solutions of the no-sign obstacle problem $\Delta w=\chi_{\{w>0\}}$ without the sign condition $w\ge0$, whose optimal regularity was settled by Andersson, Lindgren and Shahgholian. Appendix A proves that for leaky maps the free boundary is still $C^\infty$ near regular points (Theorem A.7) and that $D^3(\Pi\circ u)\in BMO$ with local boundedness near the regular part of the free boundary (Theorem A.8); but the two-dimensional optimal regularity is extended only near points satisfying a sign condition on the constraint term, because the general case requires controlling global solutions of the no-sign problem with wild behavior.

Problem Statement

The problem is to complete the fine-scale regularity program for obstacle-type constraint maps opened by the source paper. Its single scientific objective is the optimal regularity of the image map $\Pi\circ u$ and of the free boundary $\partial\{u\in M\}$, without the two restrictions under which these are currently known -- ambient dimension two, and the geometric approximation property (GAP) as an auxiliary hypothesis -- and for the full class of solutions of the Euler-Lagrange system, including non-variational (leaky) ones. Concretely, resolve each of the following five interlocking components, by proof or counterexample.

(a) GAP in $n\ge3$. Does every nonnegative solution $w$ of $\Delta w=g\chi_{\{w>0\}}$ in a bounded domain $\Omega\subset\mathbb R^n$, $n\ge2$, with positive Lipschitz $g$, admit the geometric approximation property (GAP; Definition 2.8 of the source paper, whose quantitative estimate is displayed in the Background) locally and uniformly, at every free-boundary point? In particular, prove or refute the paper's conjecture (Remark 2.9): if the singular part $\Sigma$ of the free boundary satisfies $w\in C^{2,\bar\sigma}(\Sigma\cap\bar N)$ for some neighborhood $N$ compactly contained in $\Omega$ (written $N\Subset\Omega$) and $\bar\sigma\in(0,1)$ -- that is, if the free boundary contains no anomalous points -- then $w$ verifies the GAP locally with order $2+\sigma$ uniformly in $N$, for some $\sigma(n,\bar\sigma)>0$.

(b) Optimal regularity of the image map in $n\ge3$, and GAP-independence. Is $D^3(\Pi\circ u)$ locally bounded for constraint maps in the obstacle case in every ambient dimension $n\ge2$ -- near regular free-boundary points this is known in all dimensions, and on all of $\Omega$ only in the plane (Corollary 2.7) -- at least away from the discontinuity set of the map, i.e., is $D^3(\Pi\circ u)\in L^\infty_{loc}(\Omega\setminus S)$? Equivalently, extend Theorem 2.6/Corollary 2.7 beyond $n=2$, or construct a counterexample in some $n\ge3$. This includes the paper's explicitly raised question whether the optimal $C^{2,1}$ regularity can be proved without the GAP: the published proof strongly relies on the geometric approximation property, but the validity of the $C^{2,1}$ regularity of the image map may be independent of it.

(c) Vectorial fine structure and generic regularity. Extend the scalar fine-structure theory of the singular set and the generic-regularity theory of the free boundary to the free boundary of constraint maps near its continuity points -- the paper states these developments "are widely open in this vectorial framework". The concrete targets are the following sub-claims, each independently resolvable by proof or counterexample. First, the paper's conjecture that, in ambient dimensions $n\le4$ and for generic boundary data (generic in the sense of the generic-regularity theory of Figalli, Ros-Oton and Serra, so that the exceptional set of data is of measure zero), the free boundary consists only of regular points, so that the image map is of class $C^{2,1}_{loc}$ (the dimension restriction being expected to concern only the ambient space, not the target dimension). Second, the removal, in $n\ge3$, of the exceptional set of Hausdorff dimension at most $n-3$ that currently confines the pointwise $C^{3,\sigma}$ conclusion of Theorem 2.5. Settling a sub-claim in the stated generality, by proof or counterexample, resolves that sub-claim and leaves the others open.

(d) Analyticity. Is the free boundary real-analytic around regular points in the vectorial setting? The paper's proof of $C^\infty$ regularity (via a partial hodograph-Legendre transformation and regularity for elliptic transmission problems) "does not yield the analyticity of the free boundary, which seems to be a new challenge in the vectorial setting". Settle the question: prove analyticity near regular points, or exhibit a constraint map whose free boundary is smooth but not analytic around a regular point.

(e) Leaky maps. Develop the complete two-dimensional regularity theory for leaky maps -- distributional solutions of the Euler-Lagrange system without the one-sided constraint $u(\Omega)\subset\overline M$: optimal $C^{2,1}$ regularity of the image map and regularity of the free boundary, including around points where the one-sided sign condition on the distance part $\rho\circ u$ fails. The paper's Appendix A extends the two-dimensional optimal regularity only near points satisfying a sign condition on the constraint term; the general case requires controlling global solutions of the no-sign obstacle problem with wild behavior and a much more refined analysis.

Logical relations among the components: (a) is the conjectured missing input for (b) in $n\ge3$ -- the proof of Theorem 2.6 shows that a uniform GAP for $\rho\circ u$ yields $D^3(\Pi\circ u)\in L^\infty$; the no-anomalous-points hypothesis in (a) excludes precisely the anomalous points, where the scalar fine structure breaks down in $n\ge3$; (c) would make the conclusion of (b) hold unconditionally for generic data; (d) is the next rung of regularity beyond the $C^\infty$ theory near regular points; (e) extends (b)-type conclusions from energy minimizers to the larger non-variational class.

Completion boundary: the problem is settled when each of the five components is resolved in the stated generality -- by a proof, or by an explicit counterexample satisfying all hypotheses of that component. Resolving a proper subset of the components, or resolving one under a restriction that silently narrows it (for example, only in the plane, only under the GAP, or only for a smaller solution class), settles those components and leaves the others open.

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