Open problems from Lions's 1985 memoir 'Sur les équations de Monge-Ampère': the Monge–Ampère obstacle problem, non-convex domains, viscosity solutions, first eigenvalue, uniqueness, and boundary regularity

Pierre-Louis Lions's 1985 memoir 'Sur les équations de Monge-Ampère' (Arch. Rational Mech. Anal. 89, 93–122) studied the Dirichlet problem for the real Monge–Ampère equation det(D^2u)=H(x,u,Du) — the 'quasilinear' Monge–Ampère equation, whose nonnegative right-hand side depends on the position, the solution, and its gradient — with prescribed boundary data on bounded convex domains, proving existence of generalized solutions and recording a set of open directions: an obstacle problem for the Monge–Ampère operator (least convex majorant), the Dirichlet problem on non-convex domains, a viscosity-solution framework for second-order degenerate equations with links to Hamilton–Jacobi–Bellman equations, a first eigenvalue and bifurcation theory for the Monge–Ampère operator, possible weakening of a positivity hypothesis used in dimension N≥3, uniqueness versus multiplicity for non-monotone nonlinearities, and regularity up to the boundary. The task is to resolve at least one of the open directions from this list that remain without a settled complete theory: the Dirichlet problem on non-convex domains, the weakening of the positivity hypothesis (30') for N≥3, the bifurcation part of the eigenvalue direction, and the uniqueness-or-multiplicity question for non-monotone nonlinearities. The remaining directions are recorded for context.

Background

The real Monge–Ampère equation $\det(D^2u)=f$ — where $D^2u$ denotes the Hessian matrix of second partial derivatives and $\det$ the determinant — is the prototype fully nonlinear second-order elliptic equation: it is nonlinear in the highest-order derivatives, and it is elliptic (its linearization has the right sign) only on the cone of convex functions. It arises in geometry (the Minkowski problem of prescribing Gaussian curvature on convex bodies), in optimal mass transport, and in stochastic control. For the Dirichlet problem on a bounded domain, the classical notion of weak solution is due to Aleksandrov: for a convex function $u$, the normal (multi-valued gradient) map pushes Lebesgue measure forward to the Monge–Ampère measure $\det(D^2u)\,dx$, and a generalized solution makes this measure equal to the prescribed right-hand side. Pogorelov, Cheng–Yau, Bakelman and others built the regularity theory for such generalized solutions under convexity of the domain and strict positivity of the data.

Lions's 1985 memoir (the second part of a series begun in Manuscripta Mathematica 41 (1983)) studies the Dirichlet problem

$$\det(D^2u)=H(x,u,Du)\ \text{in}\ \mathcal{O},\qquad u=\varphi\ \text{on}\ \partial\mathcal{O},$$

in a bounded convex domain $\mathcal{O}\subset\mathbb{R}^N$, where $H(x,t,p)$ is a continuous nonnegative (that is, $H\ge 0$) nonlinearity depending on the position $x$, the height $t$, and the gradient $p$, and $\varphi$ is prescribed boundary data (the model case being $\varphi=0$); the solution is required to be convex on $\overline{\mathcal{O}}$. Because the right-hand side may depend on the gradient, the memoir calls this the quasilinear Monge–Ampère equation. The paper proves existence of generalized solutions through a penalisation and approximation scheme, with an iterative method for Hamiltonians that are not monotone in the height; in dimension $N\ge 3$ the existence proof requires condition (30') of the memoir, a uniform positivity of $H$ on compact subsets, used to keep the level sets of the approximating solutions strictly convex. Under a strict-convexity assumption on the domain, and assuming a subsolution exists, uniqueness holds when $\partial H/\partial t>0$ (strict monotonicity in the height); for non-monotone $H$ the paper obtains existence only and anticipates multiplicity.

Several neighboring notions are needed for the open problems. A viscosity solution (Crandall–Lions 1983) is a continuous function that satisfies the inequality constraint whenever it is touched from above or below by smooth test functions; this notion was created for first-order Hamilton–Jacobi equations, and extending it to degenerate second-order equations — including Monge–Ampère, whose ellipticity fails off the convex cone — was an open program at the time, closely tied to Hamilton–Jacobi–Bellman equations (the dynamic-programming equations of optimal stochastic control, which are fully nonlinear and degenerate in the same way). The obstacle problem for the Laplacian asks for the smallest superharmonic function lying above a prescribed obstacle; the solution is harmonic on the set where it does not touch the obstacle. The Monge–Ampère analog seeks the least convex majorant, and it is connected to the pluripotential theory of Bedford: extremal plurisubharmonic functions (plurisubharmonic functions are the complex analogues of subharmonic functions) and envelopes of holomorphy solve degenerate complex Monge–Ampère equations and obey obstacle-type constraints. The first eigenvalue of the Laplacian on a domain is the smallest $\lambda$ for which $-\Delta v=\lambda v$ admits a nontrivial solution vanishing on the boundary; because the Monge–Ampère operator is homogeneous of degree $N$ in $D^2v$ rather than linear, the analogous 'first eigenvalue' — $\lambda_1$ such that $\det(D^2v)=\lambda_1(-v)^N$ has a nontrivial convex solution with $v\le 0$, vanishing on the boundary — has no immediate spectral theory, and its existence, uniqueness and variational characterization are nontrivial questions that interact with bifurcation of the Dirichlet problem. Finally, boundary regularity for $\det(D^2u)=f$ — classical solvability and $C^{2,\alpha}$ estimates up to the boundary on smooth strictly convex domains — was being established around 1984 by Caffarelli, Nirenberg and Spruck and by Krylov, contemporaneously with the memoir's addendum.

Problem Statement

The problem is the collection of open directions recorded by Lions's 1985 memoir on the Dirichlet problem for the real Monge–Ampère equation

$$\det(D^2u)=H(x,u,Du)\ \text{in}\ \mathcal{O},\qquad u=\varphi\ \text{on}\ \partial\mathcal{O},$$

where $\mathcal{O}\subset\mathbb{R}^N$ is a bounded convex domain, $\varphi$ is prescribed boundary data (the model case being $\varphi=0$), $H(x,t,p)$ is a continuous nonnegative nonlinearity in the position, the height and the gradient, and the solution is sought convex on $\overline{\mathcal{O}}$ (the memoir's quasilinear Monge–Ampère equation). The directions, in the source's own terms, are: (i) develop the obstacle problem for the Monge–Ampère operator, analogous to the obstacle problem for the Laplace operator — that is, characterize the least convex majorant of prescribed data that satisfies the equation below the obstacle — in connection with Bedford's work on envelopes of holomorphy; (ii) solve the equation when $\mathcal{O}$ is not convex, where the solution is not expected to be globally convex and the Aleksandrov generalized-solution theory, which is built on convex functions, does not directly apply; (iii) relate the equation to Hamilton–Jacobi–Bellman equations and construct a notion of viscosity solution for second-order degenerate equations such as Monge–Ampère, extending what Crandall–Lions did for first-order Hamilton–Jacobi equations; (iv) establish a first eigenvalue $\lambda_1$ of the Monge–Ampère operator — a value such that $\det(D^2v)=\lambda_1(-v)^N$ has a nontrivial convex solution vanishing on the boundary with $v\le 0$ inside the domain — and study bifurcation and multiplicity of solutions when $\partial H/\partial t$ exceeds $\lambda_1$; (v) for $N\ge 3$, decide whether condition (30') of the memoir — that $H$ admit positive lower bounds $\alpha_\delta>0$ with $H\ge\alpha_\delta$ on the sets $\mathcal{O}_\delta\times(-1/\delta,1/\delta)\times B_{1/\delta}$, where $\mathcal{O}_\delta$ is the set of points of $\mathcal{O}$ at distance $>\delta$ from $\partial\mathcal{O}$ and $B_{1/\delta}$ is the ball of radius $1/\delta$, used to keep the level sets of the approximating solutions strictly convex — can be replaced by a strictly weaker hypothesis; (vi) determine whether uniqueness holds when $\partial H/\partial t$ is not everywhere positive, with multiplicity examples expected; and (vii) prove regularity up to the boundary for solutions with general boundary data, going beyond the boundary estimates of Krylov, Caffarelli, Nirenberg and Spruck announced in the memoir's addendum.

These are seven distinct research questions about one boundary-value problem, recorded together because they arise from the same memoir; they are not subtasks of a single objective. The directions that remain without a settled complete theory are items (ii), (v), and (vi) together with the bifurcation part of item (iv); the other directions — (i), (iii), the eigenvalue existence and variational characterization within (iv), and (vii) — are recorded as context. The four open directions are independent: settling any one of them does not require settling the others, and their alternatives are stated within each item below.

• (ii) the Dirichlet problem on non-convex domains (where the solution is not expected to be globally convex and the Aleksandrov theory, built on convex functions, does not directly apply): either an existence theorem for $\det(D^2u)=H(x,u,Du)$, $u=\varphi$ on $\partial\mathcal{O}$, for every bounded domain $\mathcal{O}\subset\mathbb{R}^N$ with $C^{1,1}$ boundary that is not convex and every continuous boundary datum $\varphi$, with the solution concept (Aleksandrov generalized solution or viscosity solution) fixed, or a single such bounded non-convex domain with $C^{1,1}$ boundary and continuous datum for which no such solution exists;
• (iv) the bifurcation part, open only in the generality recorded because it is established on restricted classes of domains: without restricting the domain class or the nonlinearity, either a proof that a second branch of convex solutions emanates from the first eigenvalue $\lambda_1$ (bifurcation) for all bounded convex domains and all nonlinearities $H$ in the standing class, or a proof that the branch of solutions through $\lambda_1$ is locally unique in that generality;
• (v) the positivity hypothesis (30'): either a hypothesis on $H$ strictly weaker than (30') under which the Dirichlet problem admits a convex generalized solution by any argument, or a proof that no strictly weaker hypothesis suffices;
• (vi) uniqueness versus multiplicity for Hamiltonians not monotone in the height: either a uniqueness proof for the Dirichlet problem in the stated generality, or explicit examples exhibiting two distinct solutions.

References

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