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Computing multiple solutions from knowledge of the critical set

Otavio Kaminski, Diego S. Monteiro, Carlos Tomei

TL;DR

The paper develops a geometric framework to understand and compute multiple solutions of nonlinear equations by examining the global geometry of maps between Banach spaces with equal dimension and index-zero Jacobians. It introduces critical sets, tile decompositions, and the flower construction to explain how many preimages a point can have and how continuation behaves near folds and boundaries. The framework is demonstrated on visualizable plane maps, discretized nonlinear Sturm-Liouville problems, and semilinear elliptic operators, showing how spectral information and bifurcation diagrams can reveal many solutions and guide numerical searches. This work bridges geometric analysis and numerical continuation, providing practical strategies for identifying abundant preimages in both continuous and discrete settings.

Abstract

{We explore a simple {\it geometric model} for functions between spaces of the same dimension (in infinite dimensions, we require that Jacobians be Fredholm operators of index zero). The model combines standard results in analysis and topology associated with familiar global and local aspects. Functions are supposed to be proper on bounded sets. The model is valid for a large class of semilinear elliptic differential operators. It also provides a fruitful context for numerical analysis. For a function $F: X \to Y$ between real Banach spaces, continuation methods to solve $F(x) = y$ may improve from considerations about the global geometry of $F$. We consider three classes of examples. First we handle functions from the Euclidean plane to itself, for which the reasoning behind the techniques is visualizable. The second, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special right hand sides admit abundant solutions. Finally, we compute the six solutions of a semilinear elliptic equation $-Δu - f(u) = g$ studied by Solimini.}

Computing multiple solutions from knowledge of the critical set

TL;DR

The paper develops a geometric framework to understand and compute multiple solutions of nonlinear equations by examining the global geometry of maps between Banach spaces with equal dimension and index-zero Jacobians. It introduces critical sets, tile decompositions, and the flower construction to explain how many preimages a point can have and how continuation behaves near folds and boundaries. The framework is demonstrated on visualizable plane maps, discretized nonlinear Sturm-Liouville problems, and semilinear elliptic operators, showing how spectral information and bifurcation diagrams can reveal many solutions and guide numerical searches. This work bridges geometric analysis and numerical continuation, providing practical strategies for identifying abundant preimages in both continuous and discrete settings.

Abstract

{We explore a simple {\it geometric model} for functions between spaces of the same dimension (in infinite dimensions, we require that Jacobians be Fredholm operators of index zero). The model combines standard results in analysis and topology associated with familiar global and local aspects. Functions are supposed to be proper on bounded sets. The model is valid for a large class of semilinear elliptic differential operators. It also provides a fruitful context for numerical analysis. For a function between real Banach spaces, continuation methods to solve may improve from considerations about the global geometry of . We consider three classes of examples. First we handle functions from the Euclidean plane to itself, for which the reasoning behind the techniques is visualizable. The second, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special right hand sides admit abundant solutions. Finally, we compute the six solutions of a semilinear elliptic equation studied by Solimini.}
Paper Structure (18 sections, 14 theorems, 49 equations, 14 figures, 2 tables)

This paper contains 18 sections, 14 theorems, 49 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic geometry
  3. A simple function satisfying the geometric model
  4. The simplest critical points: folds
  5. Boundary points
  6. Proper functions
  7. Proof of Theorem \ref{['theo:model']}
  8. The strategy on some visualizable applications
  9. Singularities and global properties of $F$
  10. Discretized nonlinear Sturm-Liouville operators
  11. The continous case: $F_{a\ell}$ and $F_{p\ell}$
  12. The discrete case: $F^h_{a\ell}$ and $F^h_{p\ell}$
  13. Piecewise linear geometry: the critical set of $F^h_{p\ell}$
  14. Semi-linear perturbations of the Laplacian
  15. Some terminology and basic facts
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $X, Y, U, D$ as above and $F: D \to Y$ be a proper function.

Figures (14)

  • Figure 1: Five domain tiles, two image tiles.
  • Figure 2: Near a fold $u_\ast$.
  • Figure 3: Behavior at a well behaved boundary.
  • Figure 4: Pleats along fold lines $A, B, C,...$, the bifurcation diagram $\mathcal{B}$ and some preimages $P_i$ of $g$.
  • Figure 5: The critical set ${\mathcal{C}}$, the line $r$ and their images.
  • ...and 9 more figures

Theorems & Definitions (24)

  • Theorem 1.1: Geometric model
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 4.1: Costa-Figueiredo-Srikanth COSTA
  • Theorem 4.2: Teles-Tomei TELES
  • Theorem 5.1
  • Proposition 5.2
  • proof
  • ...and 14 more