Dynamics of Partial Differential Equations

Volume 19 (2022)

Number 2

A remark on the Strichartz inequality in one dimension

Pages: 163 – 175

DOI: https://dx.doi.org/10.4310/DPDE.2022.v19.n2.a4

Authors

Ryan Frier (Department of Mathematics, Kansas University, Lawrence, Ks., U.S.A.)

Shuanglin Shao (Department of Mathematics, Kansas University, Lawrence, Ks., U.S.A.)

Abstract

In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler–Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi [7] and Hundertmark–Zharnitsky [11].

Keywords

Schrödinger equations, Strichartz’s inequality, extremizers

2010 Mathematics Subject Classification

Primary 35-xx. Secondary 42-xx.

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Received 15 January 2021

Published 19 May 2022