[Paper Review] Some multi-valued solutions to Monge-Ampere equations

Luis Caffarelli, Yanyan Li|ArXiv.org|May 6, 2005

Geometry and complex manifolds3 references3 citations

TL;DR

This paper constructs multi-valued viscosity solutions to the Monge-Ampère equation in higher dimensions by leveraging covering spaces and viscosity theory. It establishes the existence of bounded, locally convex solutions on k-sheeted covers of domains with singular sets, extending classical results to multi-valued settings with precise asymptotic and boundary behavior.

ABSTRACT

We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.

Motivation & Objective

• To extend the theory of Monge-Ampère equations to multi-valued solutions in higher dimensions.
• To analyze the behavior of solutions near singular sets using covering space constructions.
• To establish existence and regularity of bounded, locally convex viscosity solutions on k-sheeted covers.
• To characterize boundary and asymptotic behavior of solutions in relation to harmonic majorants.
• To generalize classical solvability results to multi-valued settings via viscosity and Perron-type methods.

Proposed method

• Constructs a k-sheeted covering space $M_k$ over $D \setminus \Gamma$, where $\Gamma$ is a codimension-2 singular set.
• Uses Perron's method to solve the Laplace equation on $M_k$ with boundary data $\varphi_i$ on $k$ copies of $\partial D$.
• Introduces a harmonic majorant $h$ on $M_k$ that controls the solution behavior near $\Gamma$.
• Applies viscosity theory to the Monge-Ampère equation $\det(D^2u) = f$ on $M_k$ with $f$ bounded away from zero.
• Constructs subsolutions $\underline{u}$ and supersolutions $\overline{u}$ using quadratic polynomials and group actions.
• Defines the solution $u$ as the pointwise supremum of all subsolutions satisfying boundary and singularity conditions.

Experimental results

Research questions

• RQ1 Can bounded, locally convex viscosity solutions to the Monge-Ampère equation be constructed on multi-sheeted covering spaces in higher dimensions?
• RQ2 How do solutions behave near a codimension-2 singular set $\Gamma$ in the domain?
• RQ3 What is the role of harmonic majorants in controlling the asymptotic and boundary behavior of multi-valued solutions?
• RQ4 Can the viscosity solution theory be extended to non-simply connected domains with singularities via covering space techniques?
• RQ5 How do group actions and symmetries on the covering space influence the structure of solutions?

Key findings

• The Monge-Ampère equation $\det(D^2u) = f$ on $M_k$ admits at least one bounded, locally convex viscosity solution $u$ satisfying $u \leq h$, where $h$ is the harmonic majorant.
• Solutions satisfy $|u(x,m) - \bar{h}(\bar{x})| \leq C|x - \bar{x}|^\alpha$ near $\Gamma$, with $0 < \alpha < 1$, ensuring Hölder continuity up to the singular set.
• The solution $u$ is invariant under the group action $u(x,g) = u(T(g)x, \bar{g})$, reflecting the topological structure of the covering space.
• Asymptotic behavior satisfies $\limsup_{|x|\to\infty} |x| \cdot |u(x,g) - Q(T(g)x)| < \infty$ for all $g \in G$, indicating controlled growth.
• The solution $u$ achieves the boundary data $\varphi_i$ on $\partial D \times \{i\}$ and satisfies the maximum principle in the viscosity sense.
• The largest solution $u^*$ is characterized as the supremum of all viscosity subsolutions satisfying the boundary and singularity conditions.