Lions’ relaxed-positivity conjecture for the real Monge-Ampère Dirichlet problem: $W^{2,\infty}$ solutions in dimensions $N\ge 3$ under smooth degeneracy of the right-hand side

The real Monge-Ampère Dirichlet problem $\det D^2u=f$ in $O$, $u=0$ on $\partial O$, was given a purely PDE existence theory (whole-space penalization plus a Hamilton-Jacobi-Bellman reformulation) by P.-L. Lions in 1983, for smooth strictly convex domains and right-hand sides with the strict global positivity $f\ge\beta>0$. The paper’s final remarks conjecture that this uniform lower bound is unnecessary for second-derivative boundedness: $f>0$ near $\partial O$ together with $f^{1/N}\in W^{2,\infty}(O)$ should still imply $u\in W^{2,\infty}(O)$ — a statement the paper establishes only for $N=2$. The companion questions raised by the same remarks have since been settled: boundary smoothness under uniform positivity (Krylov; Caffarelli-Nirenberg-Spruck, 1984), the complex Monge-Ampère Dirichlet problem (Caffarelli-Kohn-Nirenberg-Spruck, 1985), the Hamilton-Jacobi-Bellman generalized-solution framework (the viscosity-solutions theory), and the Monge-Ampère eigenvalue (Lions, Tso, and later Le). The open core — the relaxed-positivity $W^{2,\infty}$ conjecture in dimensions $N\ge 3$ — has not been settled by later work: two-dimensional results (Guan, Guan-Sawyer, Hong-Huang-Wang, Daskalopoulos-Savin, Daskalopoulos-Lee) exploit planar algebra unavailable in higher dimensions, $N$-dimensional results (Savin 2014, Le-Savin 2017) assume power-law vanishing of $f$ near the boundary, and the known counterexamples (Wang, Mooney) use degeneracies outside the conjectured hypothesis class. The problem asks for a proof or an explicit counterexample.

Background

The real Monge-Ampère equation is the fully nonlinear second-order partial differential equation

$$\det D^2u=f(x)\quad\text{in }O,$$

where $O\subset\mathbb{R}^N$ is a bounded domain, $D^2u=(\partial_{ij}u)$ denotes the Hessian matrix of second partial derivatives, $\det D^2u$ its determinant, and $f\ge 0$ is a prescribed density. On convex solutions the equation is elliptic (linearization at $D^2u$ gives $f\,\operatorname{Tr}((D^2u)^{-1}D^2v)$), and it is the prototype of fully nonlinear degenerate elliptic equations; it prescribes the Gauss curvature of graphs, governs problems of affine geometry, and controls the regularity of optimal transport maps — the maps that rearrange one probability density into another at minimal quadratic cost, and are themselves governed by Monge-Ampère equations. The natural Dirichlet problem asks for a convex $u$ with $u=0$ on $\partial O$. Because $f$ may vanish and convex functions need not be twice differentiable, the standard solution notion is the generalized (Aleksandrov) solution: for convex $u$ the Monge-Ampère measure $\det D^2u$ is defined through the change-of-variables behavior of the gradient map $Du$, and $u$ solves the equation when this measure has density $f$.

Two regularity scales matter below. $W^{2,\infty}(O)$ is the Sobolev space of functions whose weak second derivatives are essentially bounded; for convex $u$ this is equivalent to $u\in C^{1,1}$, a Lipschitz gradient, i.e. bounded Hessian entries almost everywhere. $C^\infty(\overline O)$ means smoothness up to and including the boundary.

In “Sur les équations de Monge-Ampère. I” (Manuscripta Mathematica 41 (1983)) P.-L. Lions developed a purely PDE method — no convex-geometric constructions — for this Dirichlet problem: the problem is penalized onto the whole space, $\det(\partial_{ij}u-\frac{1}{\varepsilon}p\,u\,\delta_{ij})=f$ with $p\equiv 0$ on $\overline O$ and $p>0$ outside, and the determinant is rewritten via the algebraic identity $(\det B)^{1/N}=\inf_{A\in V}\operatorname{Tr}(AB)$ (over the set $V$ of positive symmetric matrices with $\det A=N^{-N}$) as a Hamilton-Jacobi-Bellman equation — that is, a supremum of uniformly elliptic linear operators — so that Bellman-equation regularity theory yields smooth approximants converging to the convex solution. For smooth strictly convex $O$ (boundary of everywhere positive curvature) and $f\in C_b^\infty$ with the strict global positivity $f\ge\beta>0$, Lions obtained the unique convex solution $u\in C^2(O)\cap C(\overline O)\cap C^\infty(O)\cap W^{1,\infty}(O)$; under weaker hypotheses ($f\in W^{1,\infty}(O)$, $f+\frac C2|x|^2$ convex, $f>0$ on $\overline O$) he proved the interior Hessian bound $|D^2u(x)|\le C'\operatorname{dist}(x,\partial O)^{-1}$. Boundary smoothness under strict positivity was completed immediately afterwards by two independent theories: Krylov’s boundary $C^{2,\alpha}$ estimates for fully nonlinear uniformly elliptic equations, and the Caffarelli-Nirenberg-Spruck theory of the Monge-Ampère Dirichlet problem, giving $u\in C^\infty(\overline O)$ for smooth data on smooth strictly convex domains.

The paper’s final remarks, however, single out what strict positivity buys. The author conjectures that the uniform lower bound is not needed for $W^{2,\infty}$ regularity: it should suffice that $f^{1/N}\in W^{2,\infty}(O)$ and that $f$ be positive near the boundary. The paper proves this only for $N=2$. The conjecture sits at a well-marked frontier. In the plane the identity $\det D^2u=u_{11}u_{22}-u_{12}^2$ lets any positive lower bound on the trace $\Delta u$ control both eigenvalues of $D^2u$ — exactly the mechanism of the later two-dimensional theory of Guan and of Guan-Sawyer — but no trace-to-eigenvalue mechanism exists for $N\ge 3$. On the negative side, homogeneous two-dimensional counterexamples of X.-J. Wang and the counterexamples of Mooney (in every dimension $n\ge 2$, with degeneracies whose vanishing sets have irregular support) show that without smooth control of the degeneracy the second derivatives can fail to be merely $W^{2,1}$-bounded. On the positive side, Savin and Le-Savin proved boundary $C^2$ and $C^{2,\beta}$ estimates when $f$ degenerates with a pure power law, $f\sim\operatorname{dist}(\cdot,\partial\Omega)^{\alpha}$ with $\alpha>0$. What no later work covers, in either direction, is the general class of the conjecture: smooth degeneracy of $f$ controlled only through $f^{1/N}\in W^{2,\infty}$, with arbitrary degeneracy profile, in dimensions $N\ge 3$.

Problem Statement

Let $N\ge 3$, and let $O\subset\mathbb{R}^N$ be a bounded domain with smooth, strictly convex boundary. Let $f\ge 0$ be a function on $\overline O$ satisfying the two relaxed hypotheses

• $f^{1/N}\in W^{2,\infty}(O)$, and
• $f$ is strictly positive in a neighborhood of $\partial O$ (the literal reading of the source conjecture).

Let $u$ be the unique convex generalized (Aleksandrov) solution of

$$\det D^2u=f\quad\text{in }O,\qquad u=0\quad\text{on }\partial O.$$

Prove, or disprove by an explicit counterexample, Lions’ relaxed-positivity conjecture:

$$u\in W^{2,\infty}(O),$$

that is, the second weak derivatives of $u$ are essentially bounded in $O$ (equivalently, $u$ has a Lipschitz gradient in $O$, i.e. $u\in C^{1,1}$). A valid disproof exhibits a smooth strictly convex domain $O$ and an explicit $f$ meeting the two hypotheses above with $\inf_{\overline O}f=0$, whose convex solution has second derivatives unbounded in $O$. The reference points: with the strict global positivity $f\ge\beta>0$ the conclusion holds by the classical boundary theory (Krylov; Caffarelli-Nirenberg-Spruck), while Lions’ own extension theorem gives only the blow-up bound $|D^2u(x)|\le C'\operatorname{dist}(x,\partial O)^{-1}$ in the degenerate regime — the conjecture asserts that this blow-up can be removed under the stated smoothness control of the degeneracy.

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