Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-uniqueness and symmetries for the Nirenberg problem using computer assistance

Daniel Platt

Abstract

We apply verified numerics to the Nirenberg problem, proving that a genuine solution exists near two given computer-generated approximate solutions. This proves existence of a solution for a particular prescribed curvature that was previously predicted, but not proved, to exist. We are also able to determine the symmetry groups of the genuine solutions exactly, which in one case is different from the symmetry of the prescribed curvature. We expect the computer code for this proof can be reused to study other aspects of the Nirenberg problem.

Non-uniqueness and symmetries for the Nirenberg problem using computer assistance

Abstract

We apply verified numerics to the Nirenberg problem, proving that a genuine solution exists near two given computer-generated approximate solutions. This proves existence of a solution for a particular prescribed curvature that was previously predicted, but not proved, to exist. We are also able to determine the symmetry groups of the genuine solutions exactly, which in one case is different from the symmetry of the prescribed curvature. We expect the computer code for this proof can be reused to study other aspects of the Nirenberg problem.

Paper Structure

This paper contains 19 sections, 9 theorems, 54 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The Nirenberg Problem
  4. Spherical Harmonics
  5. Geometric Analysis
  6. Rigorous computation
  7. Interval arithmetic.
  8. Quadrature.
  9. Construction of the solution
  10. Finding an approximate solution $u_0$
  11. Rigorous bounds for $u_0$
  12. Bound for the potential.
  13. Injectivity estimate for the linearised operator
  14. An eigenvalue bound
  15. A priori estimate
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $K=Y_{3,2}$ be a unit-norm spherical harmonic of bi-degree $(3,2)$. Then there exist at least five distinct solutions to equation:nirenberg-pde. One solution has a $T_d$-symmetry and the other $4$ solutions have a $S_3$-symmetry only.

Figures (1)

  • Figure 1: Left to right: $K=Y_{3,2}$, an approximate solution to \ref{['equation:nirenberg-pde']} with $S_3$-symmetry, an approximate solution to \ref{['equation:nirenberg-pde']} with $T_d$-symmetry in Mercator projection. Dashed lines show the symmetry planes of the three functions. For $Y_{3,2}$ these are genuine symmetries. The approximate solutions are not exactly invariant under these ambient symmetries. However, it is shown in the proof of \ref{['theorem:main-existence']} that genuine solutions to \ref{['equation:nirenberg-pde']} exist near the approximate solutions which have precisely the shown symmetry groups. The approximate solutions have the notation $u_0'$ in \ref{['subsection:finding-an-approximate']}.

Theorems & Definitions (18)

  • Theorem 1.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.9
  • ...and 8 more