The obstacle problem for the p-Laplacian, min{-Δ_p u, u-φ}=0, is well understood near free boundary points where the obstacle gradient is nonzero, because the operator is there uniformly elliptic and the classical theory applies. At free boundary points where ∇φ=0 the operator degenerates (p>2) or becomes singular (1<p<2), and Figalli-Krummel-Ros-Oton (J. Differential Equations, 2017) showed this regime is genuinely different: they constructed explicit 2-homogeneous global solutions in two dimensions whose free boundary has a corner at the origin, so C¹ regularity fails in general, and some profiles are non-convex, blocking the transfer of Caffarelli's convexity-based classification from the Laplacian; under a concavity-type hypothesis on φ they proved quadratic nondegeneracy and countable (n-1)-rectifiability of the free boundary, while explicitly posing the question of whether the free boundary has locally finite (n-1)-dimensional Hausdorff measure there. The open problem is to determine the fine geometric and measure-theoretic structure of the free boundary at vanishing-gradient points, namely Hausdorff finiteness under the concavity hypothesis, classification of blow-up profiles, and a regular/singular stratification.

Background

The obstacle problem asks for the smallest superharmonic function lying above a given function: given an obstacle φ\varphi, find u≥φu\ge\varphi solving min⁡{−Δpu, u−φ}=0\min\{-\Delta_p u,\,u-\varphi\}=0 in a domain Ω⊂Rn\Omega\subset\mathbb{R}^n, where Δpu=div⁡(∣∇u∣p−2∇u)\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u) is the pp-Laplacian. The solution is pp-harmonic (Δpu=0\Delta_p u=0) in the non-coincidence set {u>φ}\{u>\varphi\} and coincides with the obstacle on the contact set {u=φ}\{u=\varphi\}; the interface ∂{u>φ}\partial\{u>\varphi\} between them is the free boundary. For p=2p=2 this is the classical obstacle problem, whose free-boundary theory is a cornerstone of nonlinear analysis: Caffarelli established a quadratic nondegeneracy of u−φu-\varphi at free boundary points (the solution cannot detach from the obstacle faster than quadratically), that the free boundary has locally finite (n−1)(n-1)-dimensional Hausdorff measure Hn−1\mathcal{H}^{n-1} (the standard surface measure; countable (n−1)(n-1)-rectifiability means the set is covered, up to an Hn−1\mathcal{H}^{n-1}-null set, by countably many Lipschitz images of Rn−1\mathbb{R}^{n-1}), a decomposition of free boundary points into regular points — where the free boundary is locally a C1,αC^{1,\alpha} hypersurface — and singular points, and a classification of the blow-up profiles, i.e. the possible limits of rescalings of u−φu-\varphi at a free boundary point, which for the Laplacian are exactly the 2-homogeneous global solutions, all convex and of the form c (x1+)2c\,(x_1^+)^2 up to rotations.

For p≠2p\neq2 the operator degenerates: Δp\Delta_p is uniformly elliptic only where ∇u≠0\nabla u\neq0 (degenerate at critical points when p>2p>2, singular when 1<p<21<p<2). Andersson, Lindgren and Shahgholian proved the optimal C1,αC^{1,\alpha} regularity of solutions near free boundary points. Near a free boundary point x0x_0 with ∇φ(x0)≠0\nabla\varphi(x_0)\neq0 one also has ∇u(x0)≠0\nabla u(x_0)\neq0, so the operator is uniformly elliptic there and the classical theory carries over. The genuinely open regime, isolated by Figalli, Krummel and Ros-Oton, consists of the free boundary points where ∇φ=0\nabla\varphi=0 (critical points of the obstacle): there the degeneracy of the operator coincides with the contact, and prior to their work nothing was known. They proved three things. First, for every p≠2p\neq2 there exist explicit global 2-homogeneous solutions in R2\mathbb{R}^2 with obstacle φ=−∣x∣2\varphi=-|x|^2 whose non-coincidence set is a cone of angle θ0=2kπ(1−p−12p)≠π,k∈{1,2,3},\theta_0=2k\pi\left(1-\sqrt{\tfrac{p-1}{2p}}\right)\neq\pi,\qquad k\in\{1,2,3\}, so the free boundary has a corner at the origin and C1C^1 regularity cannot hold in general at vanishing-gradient points; moreover, unlike the Laplacian case, some of these global solutions are not convex, which the authors identify as a major obstacle to transferring Caffarelli's classification strategy. Second, under the concavity-type hypothesis ∣∇φ∣2−p Δpφ≤−c0<0in {∇φ≠0},|\nabla\varphi|^{2-p}\,\Delta_p\varphi\le-c_0<0\quad\text{in }\{\nabla\varphi\neq0\}, they proved a quadratic nondegeneracy of u−φu-\varphi at every free boundary point (their Theorem 1.5). Third, under the same hypothesis the free boundary is countably (n−1)(n-1)-rectifiable (their Theorem 1.8). However, their Remark 3.3 records that these arguments give local finiteness of Hn−1\mathcal{H}^{n-1} only away from the degenerate points and cannot conclude Hn−1(∂{u>φ}∩K)<∞\mathcal{H}^{n-1}(\partial\{u>\varphi\}\cap K)<\infty for compact KK, because the free-boundary portion Γ1\Gamma_1 may be badly behaved near points where ∇φ=0\nabla\varphi=0.

This should be contrasted with the formally similar reaction-type problem Δpu=f χ{u>0}\Delta_p u=f\,\chi_{\{u>0\}} with zero obstacle (often called the p-obstacle problem), for which local finiteness of Hn−1\mathcal{H}^{n-1} of the free boundary is known — Lee and Shahgholian for 2<p<∞2<p<\infty, extended to quasilinear operators by Challal, Lyaghfouri, Rodrigues and Teymurazyan, with almost-everywhere regularity of the free boundary for the normalized problem by Andersson. For p≠2p\neq2 the two problem classes are of genuinely different nature, and those methods do not transfer to the obstacle problem with a nonconstant obstacle precisely at points where ∇φ=0\nabla\varphi=0.

Problem Statement

Let 1<p<∞1<p<\infty with p≠2p\neq2, let Ω⊂Rn\Omega\subset\mathbb{R}^n, let φ∈C2(Ω)\varphi\in C^2(\Omega), and let uu solve the pp-Laplacian obstacle problem min⁡{−Δpu, u−φ}=0in Ω,\min\{-\Delta_p u,\,u-\varphi\}=0\quad\text{in }\Omega, where Δpu=div⁡(∣∇u∣p−2∇u)\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u) is the pp-Laplacian and Hk\mathcal{H}^{k} denotes kk-dimensional Hausdorff measure. Determine the fine geometric and measure-theoretic structure of the free boundary ∂{u>φ}\partial\{u>\varphi\} near free boundary points x0x_0 with ∇φ(x0)=0\nabla\varphi(x_0)=0, to the level of completeness of the classical p=2p=2 theory. Concretely, the problem consists of three linked layers, each of which is open:

  1. Hausdorff finiteness — the question posed explicitly in the source. Under the concavity-type hypothesis ∣∇φ∣2−pΔpφ≤−c0<0|\nabla\varphi|^{2-p}\Delta_p\varphi\le-c_0<0 on {∇φ≠0}\{\nabla\varphi\neq0\}, under which countable (n−1)(n-1)-rectifiability of the free boundary (coverage up to an Hn−1\mathcal{H}^{n-1}-null set by countably many Lipschitz images of Rn−1\mathbb{R}^{n-1}) is already known, decide whether Hn−1(∂{u>φ}∩K)<∞\mathcal{H}^{n-1}(\partial\{u>\varphi\}\cap K)<\infty for every compact K⊂⊂ΩK\subset\subset\Omega. Either an affirmative proof or an explicit counterexample (a solution/obstacle pair satisfying the hypothesis whose free boundary has infinite Hn−1\mathcal{H}^{n-1}-measure on some compact set) answers this question. Without any concavity hypothesis, even rectifiability at vanishing-gradient points is unknown.

  2. Blow-up classification. Classify the homogeneous global solutions that arise as blow-up profiles of u−φu-\varphi at vanishing-gradient free boundary points (a blow-up profile being the limit of rescalings of u−φu-\varphi centered at the point as the scale tends to zero, whenever such a limit exists), for general p≠2p\neq2 and n≥2n\ge2 — beyond the family of explicit two-dimensional 2-homogeneous solutions with cone angles θ0=2kπ(1−(p−1)/(2p))\theta_0=2k\pi\left(1-\sqrt{(p-1)/(2p)}\right) constructed in the source. Since some of these profiles are non-convex, the classification cannot follow Caffarelli's convexity-based scheme for the Laplacian and requires a new mechanism.

  3. Stratification and regularity. Stratify the vanishing-gradient free boundary points into regular and singular classes with a geometric characterization of each class, and determine the regularity of the regular part — in particular, which subsets of the free boundary are C1C^1 or smoother near such points. Theorem 1.1 of the source shows that C1C^1 regularity cannot hold at all vanishing-gradient points (corners exist), so the target is a Caffarelli-type decomposition rather than uniform smoothness.

These three layers are the successive standard levels of free-boundary structure theory — measure-theoretic size, blow-up classification, regularity of the regular part — and jointly constitute the complete geometric and measure-theoretic understanding of the free boundary in the degenerate/singular regime that the source identifies as remaining. They are scientifically linked: a classification feeds the stratification, and nondegeneracy/density inputs feed the measure estimate.

References

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