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A scattering operator for some nonlinear elliptic equations

Raphaël Côte, Camille Laurent

Abstract

We consider non linear elliptic equations of the form $Δu = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a \emph{one-to-one correspondence} between the non linear solution $u$ defined there, and the linear solution $u\_L$ to the Laplace equation, such that, in an adequate space, $u - u\_L\to 0$ as $|x|\to +\infty$. This is a kind of scattering operator.Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the $H$-system. Similar results are derived for solution defined on the neighborhood of a point in $\mathbb{R}^d$. The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction).

A scattering operator for some nonlinear elliptic equations

Abstract

We consider non linear elliptic equations of the form for suitable analytic nonlinearity , in the vinicity of infinity in , that is on the complement of a compact set.We show that there is a \emph{one-to-one correspondence} between the non linear solution defined there, and the linear solution to the Laplace equation, such that, in an adequate space, as . This is a kind of scattering operator.Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the -system. Similar results are derived for solution defined on the neighborhood of a point in . The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction).
Paper Structure (36 sections, 69 theorems, 539 equations)

This paper contains 36 sections, 69 theorems, 539 equations.

Table of Contents

  1. Introduction
  2. Motivations and setting of the problem
  3. The functional setting
  4. Main results on the semilinear equation
  5. Main results on conformal equations in dimension 2
  6. Main results on Harmonic maps in dimension d >=3
  7. Main results on semilinear equations close to a point
  8. Acknowledgments
  9. The linear flow and Duhamel formulation
  10. The $Y_{s,t}$ spaces
  11. The linear flow
  12. Bounds on the Duhamel term
  13. Some properties of Yst and Zst spaces
  14. Product law in the Ys,t spaces
  15. The elliptic null condition in dimension 2
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $d\geqslant 3$. Assume that $f$ as in eq:f_semilinear satisfies the supercriticality assumption Let $u_{0} \in H^s( \ifdefequal{S}{1} {\mathbbm{S}} {\mathbb{S}} ^{d-1})$, and $u_{L} \in \mathcal{Z}^{\infty}_{s}$ given in eq:ell_lin_intro. Then, there exist $r_{0}\geqslant 1$ and a unique small solution $u \in \mathcal{Z}^{\infty}_{s,r_{0}}$ of eq:ell on $\{|x|\geqslant r_{0}\}$ and such that

Theorems & Definitions (148)

  • Theorem 1.1
  • Theorem 1.2: Semilinear energy critical equation
  • Corollary 1.3
  • Theorem 1.4: Semilinear equation with decay
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 138 more