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Hyperbolic Monge-Ampère Equation on a Cylinder: Well-Posedness and Stability

Maria Deliyianni, Shankar C. Venkataramani

TL;DR

The paper develops a rigorous framework for the hyperbolic Monge–Ampère equation on cylinder-like domains by enforcing rigidity through partial convexity and employing a hodograph transformation to obtain a linear damped wave equation. It introduces hodograph weak solutions via a parametrix–corrector decomposition to resolve corner singularities and recasts the problem as a singular Volterra integral equation, establishing existence and uniqueness along with boundary data attainment. Energy methods yield quantitative stability estimates under perturbations of the curvature function \(\lambda(y)\), highlighting the forward-time well-posedness and an intrinsic arrow of time. Collectively, the work advances a rigorous, geometry-informed approach to the rigidity–flexibility spectrum in hyperbolic isometric immersion and provides a foundation for further analytic and numerical exploration.

Abstract

This paper develops a rigorous analytic framework for the hyperbolic Monge-Ampère equation on strip-like domains, which model wrinkled patterns in thin elastic sheets. Our work addresses the rigid side of the classical rigidity-flexibility dichotomy by defining this regime not by high smoothness, but by the more fundamental property of partial convexity. The hodograph transformation is the natural tool for this setting, as its validity is predicated on partial convexity. It converts the nonlinear Monge-Ampère equation into a linear damped wave equation, allowing us to formulate a well-posed Cauchy-Goursat problem. A key challenge is the corner singularity that arises where characteristic and non-characteristic boundary data meet. To resolve this, we develop a parametrix-corrector decomposition that captures the solution's inherent singular behavior. This method recasts the problem as a singular Volterra integral equation, for which we prove the existence and uniqueness of a new class of hodograph weak solutions. Finally, we derive energy estimates to establish the quantitative stability of these rigid solutions under perturbations of the underlying curvature function.

Hyperbolic Monge-Ampère Equation on a Cylinder: Well-Posedness and Stability

TL;DR

The paper develops a rigorous framework for the hyperbolic Monge–Ampère equation on cylinder-like domains by enforcing rigidity through partial convexity and employing a hodograph transformation to obtain a linear damped wave equation. It introduces hodograph weak solutions via a parametrix–corrector decomposition to resolve corner singularities and recasts the problem as a singular Volterra integral equation, establishing existence and uniqueness along with boundary data attainment. Energy methods yield quantitative stability estimates under perturbations of the curvature function \(\lambda(y)\), highlighting the forward-time well-posedness and an intrinsic arrow of time. Collectively, the work advances a rigorous, geometry-informed approach to the rigidity–flexibility spectrum in hyperbolic isometric immersion and provides a foundation for further analytic and numerical exploration.

Abstract

This paper develops a rigorous analytic framework for the hyperbolic Monge-Ampère equation on strip-like domains, which model wrinkled patterns in thin elastic sheets. Our work addresses the rigid side of the classical rigidity-flexibility dichotomy by defining this regime not by high smoothness, but by the more fundamental property of partial convexity. The hodograph transformation is the natural tool for this setting, as its validity is predicated on partial convexity. It converts the nonlinear Monge-Ampère equation into a linear damped wave equation, allowing us to formulate a well-posed Cauchy-Goursat problem. A key challenge is the corner singularity that arises where characteristic and non-characteristic boundary data meet. To resolve this, we develop a parametrix-corrector decomposition that captures the solution's inherent singular behavior. This method recasts the problem as a singular Volterra integral equation, for which we prove the existence and uniqueness of a new class of hodograph weak solutions. Finally, we derive energy estimates to establish the quantitative stability of these rigid solutions under perturbations of the underlying curvature function.

Paper Structure

This paper contains 23 sections, 15 theorems, 167 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Rigidity
  3. Geometric Setting and Problem Formulation
  4. Classical Solutions and Rigidity
  5. The Hodograph Formulation
  6. Exterior Differential System and Characteristics
  7. The Linear PDE from the Hodograph Transformation
  8. Non-integrability and the Goursat Problem
  9. Symmetric Solutions and Problem Formulation
  10. Symmetries and Inflection Lines
  11. Smooth Gluing of Symmetric Solutions
  12. Implications for the Goursat and Cauchy Problems
  13. Problem Formulation
  14. Existence and Uniqueness of Hodograph Weak Solutions
  15. Singularities and Their Resolution
  16. ...and 8 more sections

Key Result

Lemma 3.1

The characteristic subsystems csystem1 and csystem2 are Darboux integrable if and only if the function $\lambda(u)$ is constant.

Figures (2)

  • Figure 1: Geometry of the Cauchy--Goursat problem in rotated characteristic coordinates $(\sigma,\tau)$. The shaded region shows the domain of dependence for $(s,t)$.
  • Figure 2: Integration contours for Cases I–IV. The known boundary is in blue.

Theorems & Definitions (40)

  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2: $C^\infty$ Gluing
  • proof
  • Definition 5.1: Weak compatibility for the Cauchy--Goursat data
  • Definition 5.2: Attainment of boundary conditions
  • Theorem 5.3
  • proof
  • ...and 30 more