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On the fractional Schödinger equation with variable coefficients

C. E. Kenig, D. Pilod, G. Ponce, L. Vega

Abstract

We study the initial value problem (IVP) associated to the semi-linear fractional Schödinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients

On the fractional Schödinger equation with variable coefficients

Abstract

We study the initial value problem (IVP) associated to the semi-linear fractional Schödinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients

Paper Structure

This paper contains 15 sections, 27 theorems, 324 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Spectral Theory for $L=L(x)$
  4. Proof of Theorem \ref{['2.1']} in the case $\alpha=1$
  5. Proof of Theorem \ref{['2.1']} in the case $0<\alpha<1$
  6. Extension of Theorem \ref{['2.1']} to the values $\alpha >1$
  7. Sketch of different proof of the estimate \ref{['r3']}.
  8. Estimates for the non-linear results
  9. Proof of Theorem \ref{['A1']} and Theorem \ref{['A1a']}
  10. Proof of Theorems \ref{['A2']}
  11. Extension to a local problem in the upper-half space.
  12. Unique continuation for the extension problem
  13. Carleman estimate.
  14. Doubling estimate.
  15. Proof of Proposition \ref{['WUCP:ext']}

Key Result

Theorem 1.1

Let $\alpha \in (0,1)$. Assume hyp1-hyp3, asymp-hyp-P, and J where $J\in \mathbb N$, $J\geq 2\kappa-1$, for some $\kappa\in \mathbb N$ with $2\kappa>n/2$. Then for any $u_0\in H^s(\mathbb R^n)$, $s\in (n/2,2\kappa]$, there exist $T=T(\|u_0\|_{s,2};n; N_1;N_2)>0$ and a unique strong solution $u=u(x,t Moreover the (locally defined) map data $\to$ solution is analytic from $H^{s}(\mathbb R^n)$ to $X_

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 43 more